JPL PUBLICATION 77-1 5

7 ransition and Laminar Instability

(XESA-CP-153203) TSAh'SiTION AN3 LBHINAB N77 - 240 58 L S S T L B I L I T Y (Jet Propuls ion Lab.) 84 p BC kOS/flF ]!I1 'JSCL ?1A

Unclas G3/02 253226

National Aeronautics and Space Administration

Jet Propulsion Laboratory California Institute of Technology Pasadena, California 91 103

https://ntrs.nasa.gov/search.jsp?R=19770017114 2020-03-26T08:15:09+00:00Z

JPL PUBLICATlOi\j 77-1 5

Transition and Laminar Instability

Leslie M. Mack

May 15, 1977

National Aeronautics and Space Administration

Jet Propulsion Laboratory California Institute of 1 echnology Pasadena, Caiifornia 9 1 103

Preoared Under Contract No. NAS 7-100 National Aeronautics and Space Administration

PREFACE

The work described in this report w a s performed bv the Earth and Space

Sciences Division of the Jet Propulsion Laboratory.

This report is Chapter 3 of Application and Fundamentals of Turhulence,

to be published by Plencm Press of London and New York. The report was p r p -

sented as a Short Course Lecture at the University of Tennessee Space Institute,

Tullahoma, Te;messee, on January 11, 197:.

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ACKNOWLEDCEMEKT

Support from Langley Research Center is gratefully acknowledged.

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TABLE OF CONTENTS

I. Historical Background

11. S t a b i l i t y Theory

A . Formulation of t he Eigenvalue Problem

B. Temporal and S p a t i a l Theory

C . Numerical Procedures

D. Sone Numerical R e s u l t s

111. S t a b i l i t y Experiments

A . Schubauer-Skramstad Experiment

B. Other Older Experiments

C . Three Recent Experiments

IV . Trans it ion Predic t i on

A . Nature of the Problem

B . Amplitude Density Methods

C . Amplitude Method

D . Ef fec t of Freestream Turbulence on Trans i t ,>n

References

TABLES

I.

11.

Operation of Eigenvalue Search Procedure

P rope r t i e s of Falkner-Skan Bour,dary Layers

111. Comparison of S p a t i a l and Temporal Theories

f o r Three Falkner-Skan Boundary Layers

I V . S p a t i a l Amplification Rates of Oblique Waves

i n the Flat-Plate Boundary Laycr f o r Different

Values of T

1

4

4

14

2 1

27

34

34

36

39

44

44

45

49

53

60

h i

58

69

i o

-.- I 1- i i

FIGURES

1.

2 .

3.

4 .

5.

6 .

7 .

8.

I).

Envelope cunws of amplitude ratio for

Falkner-Skan boundiiry lavers

Frequency of amplitude-ratio envelope curves

for Falkncr-Skan boundary lnycrs

Frequency-response curves of amplitude ratio

at several Reynolds numbers for flat-plate

boundary layer

Reynolds number dependence of bandwidth

of frequency-response curves for Falkner-Skan

boundary layers

Direction of group velocity for oblique waves

of constant wave number in flat-plate hnundnry

layer

Comparison of theory with Schubauer-Skramstad

measurements of the growth of six constant-

frequency waves in flat-plate boundary layer

Effect of €repstream turhulence on the

tracsition Reynolds number of the flat-plate

boundary layer

One- and two-dimensional interpolation pressure

spectra of isotropic tiirbulence

Disturbance amplitiidc growth in flat-plate

boundary layer n r c o r d inq t o amplitude mrthnd

f o r several freestream turbulence levels

7 1

72

72

7 1

7 3

7 4

75

76

vi

I I

77-1 5

ABSTRACT

A review is given of the app l i ca t ion of l i n e a r s t a b i l i t y theory til

the problem of bound,ii-vr-layer t ransi t i t :> i n incompressihle i l ' w . The theclr \'

i s put i n t o a form s u i t a b l e f o r three-dimensional boundary lavers ; both the

temporal and s p a t i a l tlirsories .Ire examined; and :i g e n e r a l i z e ~ l G l s t e r rc. ln-

t i o n f o r three-dimensional boundary layers i s der ived. Numerical example5

include the s t a b i l i t y character i .s t ics of Fall.-ner-Skan boiindary l aye r s , t h e

accuracy of the two-dimensional Gaster r e l a t i o n ioi- these boundary l aye r s ,

and t h e magnitude and d i r e c t i o n of t h e group ve loc i ty f o r obl ique waves i n

the Blasius boundary layer. A review i s given of the ava i lzb le experiments

which bear on t h e v a l i d i t y of s t a b i l i t y theory and L t s r e l a t i o n t o trans1t. ion.

The f i n a l s ec t ion i s delroted t o the app l i ca t ion of s t a b i l i t y thenr;: t o t i - . T ? -

s i t i o n predic t ion . Lii.nm-lnn's method. t h e e method, and the modified I - '

method, where n is r e l a t ed t o t h e ex te rna l dis turbance l e v e l , are a l l dis-

cussed. A d i f f e r e n t t y p e ~ - , f method, c a l l e d the .-lmplituc!cl method, is deszribed

i n which t h e wide-hand dis turbance amplitude i n the boundarv l ave r is esti-

m t e d from s t a b i l i t y theory and an in t e rac t ion r e l a t i o n f o r t h e init.ici1 ai.,iiJLi-

tude dens i ty of t h e nos t unstable frequency. This mL:!lL,c: is appl ied t o thr

e f f e c t of freestream turhiilcnce on t h c t r a n s i t i n n of F7lkner-Sknn horindnry

layers .

n n

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I . HISTORICAL BACKGROUND

The earliest explanation f o r t h e appearance of turbulencc was t h a t the

laminar flow becomes unstable , and the l i n e a r s t a b i l i t y theory w a s f irst

developed t o explore t h i s p o s s i b i l i t y . A series of papers by Rayleigh

provided many notable r e s u l t s concerning the i n s t a b i l i t y of inv isc id flows,

(1)

such as i n f l e c t i o n a l i n s t a b i l i t y , but l i t t ls progress w a s made toward the

o r i g i n a l goal.

flow, but i n 1921 Prandtl") showed tha t v i s c o s i t y can a l s o be des t ab i l i z ing .

Viscosity w a s commonly thought t o a c t only t o s t a b i l i z e the

It was t h i s discovery t h a t f i n a l l y provided a mechanism f o r t he i n s t a b i l i t y

of boundary l aye r s i n zero and favorable pressure grad ien ts which a r e s t a b l e

t o purely inv isc id disturbances. However, it w a s not u n t i l sme years later

t h a t T ~ l l s i e n ' ~ ) worked out a complete theory of boundary-layer s t a b i l i t y ,

and f o r t he f i r s t time computed a meaningful c r i t i c a l Reynolds number (Recr) ,

1.e. the lowest Reynolds number a t which i n s t a b i l i t y appears.

t ha t i n s t a b i l i t y and t r a n s i t i o n t o turbulence a r e synonynous i n boundary

layers w a s dashed by the low value of Recr f o r the f l a t - p l a t e boundary layer .

Tollmien's ca lcu la t ion gave a value of 420 f o r the c r i t i c a l displacement-

thickness Reynolds number, which is equivalent t o R e

Even i n the high turbulence l e v e l wind tunnels of t h a t t i m e , t r a n s i t i o n

Any expectat ion

* * * 4 = Ulx / v = 6 x 1 0 . c r

was found

In

theory t o

5 6 between Re = 3.5 x 10 and 1 x 10 . what can be considered the earliest appl ica t ion of l i n e a r s t a b i l i t y

t r a n s i t i o n predic t ion , Schlichting") ca lcu la ted the amplitude

t

r a t i o A/A0 of the most amplified frequency as a funct ion of Reynolds numher

f o r a f la t -p lace boundary layer , and found tha t t h i s quant i ty had values between

f i v e and nine a t t h e observed t r a n s i t i o n Reynolds numbers.

Germany, the s t a b i l i t y theory received l i t t l e acceptance because of f a i l u r e t o

O u t s i d e of

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observe the predicted waves, mathematical d f f i c u l t i e s , and a l s o the f ee l ing

t h a t a l i n e a r theory could not have much t o say about t he o r i g i n of tu:rbulence

which is inherent ly non-linear. The experiment of Schubauer and Skramstad (5)

completely revised t h i s opinion, and unequivocally demonstrated the ex is tence

of i n s t a b i l i t y waves i n a boundary l aye r , t h e i r connection with t r a n s i t i o n , and

the quen t i t a t lve descr ip t ion of t h e i r behavior by the theory of Tollmien and

Schlichting.

publ icat ion, and by its very completeness seemed t o answer most quest ions

This experiment made an enormous impact a t t he time of its

concerning the l i n e a r theory. To a large exten t , subsequent experimental

work on t r a n s i t i o n went i n o ther d i r ec t ions , and the p o s s i b i l i t y t h a t l i n e a r

theory can be quan t i t a t ive ly r e l a t e d t o t r a n s i t i o n has not received a dec is ive

experimental test.

parameters such as pressure grad ien t , suc t ion and hea t t r a n s f e r q u a l i t a t i v e l y

a f f e c t t r a n s i t i o n i n the manner predicted by s t a b i l i t y theory, and i n

p a r t i c u l a r t h a t a flow predicted t o be s t a b l e by t h e theory should remain

jl- e- On t he o ther hand, i t is general ly accepted t h a t f h w

laminar. This expc-ctation has of ten been deceived. A good introduct ion

i n t o the complexity of t r a n s i t i o n and t h e d i f f i c u l t i e s involved in t ry ing

t o arrive a t a r a t i o n a l approach t o its predict ion can be found i n a repor t

by Morkovin. ( 6 )

Inves t iga tors i n Germany applied the s t ? 0 4 1 f t y theory t o boundary layers

with pressure grad ien ts and suct ion, and t h i s work is summarized tn Schl ich t ing ' s

book. (7)

the only la rge body of numerical r e s u l t s f o r exact boundary-layer so lu t ions

before the advent of the computerage by ca lcu la t ina the s t a b i l i t y charac te r i s -

tics of the Falkner-Skan family of ve loc i ty p r o f i l e s .

asymptotic theory were put on a firmer foundation by Lin,") and t h i s work has

We may make p a r t i c u l a r mention of Pretsch's(8) work, as he provided

The mather.atics of the

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When i n about 1960 t h e d i g i t a l computer reached (10) been continued by Keid.

a s t age of development permit t ing the d i r e c t numerical so lu t ion of t h e primary

d i f f e r e n t i a l equat ions, the l i n e a r theory was ext2nded t o many mre boundary-

l aye r flows:

boundary layers (Kurtz and Crandall (I2) and Nachtsheim

boundary layers (Brown (I4) and Mack ): boundary l aye r s on compliant w a l l s

(Landahl and Kaplan (16) ) ; a recomputation of t h e Falkner-Skan flows (Wazzan,

three-dimensional boundary l aye r s (Brown''')) : f ree-convection

(13)). , compressible

(15)

Okamura, and Smith(17)) ; a quasi-steady c a l c u l a t i a i of unsteady boundary l aye r s L

(Obremski, Morkovin, and Landahl ('*I) : and heated-wall water boundary l aye r s

(Wazzan, Okamura, and Smith (19)).

It w i l l be the main purpose of t h i s chapt r t o explain i n d e t a i l t h e

use of l inear s t a b i l i t y theory as a means of t r a n s i t i o n predict ion. Enough

of t he theory is presented i n Section I1 t o make i t c l e a r how the e s s e n t i a l

quant i ty , t he amplitude r a t i o A/Ao, is obtained.

s p a t i a l ampl i f ica t ion theo r i e s is discussed, and a numerical procedure presented

which allows eigenvalues t o be ca lcu la ted t o a r b i t r a r i l y high Reynolds numbers.

A few numerical examples a rc given, and i n Section I11 t h e ava i lab le experiments

bearing on s t a b i l i t y theory a re examined. F ina l ly , i n Sect ion I V t he appl ica-

t i on of l i n e a r theory t o t r a n s i t i o n predic t ion is taken up, and t h e e ,

modified e and amplitude mct i iods a r e discussed and applied t o the e f f e c t of

f reestream turbulence on t zans i t i on . It must be emphasized tha t the subjec t

matter is r e s t r i c t e d t o incompressible boundary layers along an impermeable

sur face of zero curvature i n t h e absence of body forces . Gor t le r i n s t a b i l i t y ,

f r e e shear flowS, and s t r a t i f i e d , r o t a t i n g and compressible flows, a r e a l l

excluded.

The use of temporal and

9

9

Before proceeding f u r t h c r , i t i s wcll t o vention some general references.

(21) These are review a r t i c l e s on s t a b i l i t y theory bv Schlichting"'), Shen ,

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( 2 5 ) Stuart (22) and Reid(23); and books by Lin (24) and Betchov and Criminale.

Schli~hting*s(~) book on boundary-layer theory contains two chapters on stability

and transition, and Monin and Yaglom's (26) book on turbul mce contains a sing1 ci

lengthy chapter on the same subject, as does the book by White (27) on viscous

flcw theory. (29) Reviews of transition have been given by ilryden, (28) Tani.

Morkovin") and Reshotko . (30) An extensive discussion of both stability theory

and transition, not all at high speeds in spite of the title, may be found in

(31) the Morkovin-Mack recorded lectures.

11. STABILITY THEORY

A. Formulation of Eigenvalue Problem

1. grivation of Equations

The stability theory starts with the time dependc9t Navier-Stokes

equations, not the boundary-layer equations. We will restrict ourselves to

the flow of a single incompressible fluid on a surface of negligible curvature.

This simplification eliminates many possible sources of instability, but

preserves the twdhich are essential t o an understanding of the subject:

inflectional and viscous instability. The Navier-Stokes equations for a

viscous incompressible fluid tn Cartesian coordinates are

- + ~ - + v - + w - - o au au au au - L * + v v 2 u,

av+ u-+ v - + w - = - L*+ vv v,

at ax aY az P ax

av av av 2 at ax 3Y az F aY

3w aw aw aw L - + u - + v - + w - z - a + 'Jv W, at ax aY az P az

au av aw ax ;ly a2 - + - + - - I o .

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The x-axis is in the direction of the freestream velocity, the y axis is

norm411 to the surface, and the e axis is normal to the x v r l y axes.

velocities u, v, w are in the x, y, e directions, respectively. The density

is p , the pressure p, and the kinematic viscosity cDefficient u = pip.

first three equations are the x, z, z momentum equations; the fourth equation

1 9 the equation of continuity.

The

'me

All flow quantities are divided int.0 a steady mean-flow term and au

unsteady fluctuation term. A typical term is

The mean-flow terms satisfy the boundary-layer equations.

similar to (5) are substituted into ( 1 ) - ( 4 ) for all flow variables. + ' e mean-

flow terms dropped which are negligible by the boundary-layer e

mean boundary-layer equations subtracted out, and the nonlinear terms neglected,

a much simplified, but still too coL,Jlicated, system of equati dll: results. The

additional assumpticn of locally ptiralli-. f Tow,

When expressions

Vt ions * the

reduces the equations sufficiently so that upon the introduction of a sinus;idal

disturbance they become ordinary differential equations.

The parallel-flow Equations aro,, in dimensionless form,

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From now until Section IV dimensional quan-ities will be denoted by asterisks. * *

The velocity scale is the freestream velocity U the length scale is L (left 1' - * *' unspecified for the present), and the pressure scale nu1 . number is

The Reynolds

* * f R = U I L /v . (11)

These equations, which are the basis of almost all stability investigations,

are exact for the flow in a channel or for Couette flow, but are only an

approximation for boundary-layer flows.

method first applied to this problem by Bouthier (32) and later by Saric and

Nayfeh(33)is used, the above equations appear as the zeroth approximation.

The nsxt approximation takes the growth of the boundary layer into account.

Although this method olay be desirable for refined calculations, it is not

needed for an elrrnentary presentation of the subject and will not be pursued

here.

2. Introduction of Sinusoidal Disturbances

If the multiple scale or two-timing

The final form of the differential equations, where the coefficients

are functions only of y, and x, z , t appear only as derivatives, suggests

the following type of disturbance:

77-15

Rera, f (y ) , O(y), h(y), w(y) are t h e complex amplitude funct ions of the

d i s t u r b m c e flow va r i ab le s u' . 9' , w' , p' ; a and B ere the dfmensionless wave

numbers 2nL /Ax and 2nL / A z , where Ax aud X are the wavelewths i n the x and

t d i rec t ions , respect ively, and w is t h e dimensionless frequency o L /Ul.

t h e moment, a, B, o may be e i t h e r real or complex.

* * * * * * z * * *

For

When (12) is subs t i t u t ed i n t o (7)-(10), t he following equations for

t h e amplitude function are obtained:

1 2 2 i(au + BW-W) f + V'Q = - i a n + E [ f*' - (a + B ) f l ,

2 2 + B i(au + OW-@) h + W*Q = - iBn + 1 [h" - (a hl ,

The primes now r e f e r t o d i f f e r e n t i a t i o n with respect to y.

conditions are t h a t t he no-slip condition app l i e s a t t h e w a l l ,

The boundary

f(0) = 0, Q ( 0 ) = 0, h(O) = 0 ,

and t h a t t he disturbances go t o zero (or are a t least bounded) as y + a,

Since a l l of t h e boundary conditions are homogeneous, it can be expected t h a t

so lu t ions t o (13)-(16) w i l l e x i s t only for p a r t i c u l a r combinations of R, a , 8 ,

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and w.

evaluate the eigenvalue relation

Consequently, we have an eigenvalue problem, and the primary tabk is to

fcr one parameter in terms of the others.

3. Orr-Soaanerfeld Equation

If (13) and (15) are combined to form af + Bh, this combination can be

eliminated by (16), and, after differentiation, T' can be eliminated by (14)

to give

When W = 0, this equation reduces to the equation obtained by Squire, ( 3 4 ) and

when in addition 0 5 0, to

2 'I 4 $iv - 2a 4 + a 4

This is the Orr-Somerfeld equation and is the basis for most of the work done

in incomprwsible stability theory.

The Orr-Sonanerfeld equation is a fourth-order equation and applies to a

two-dimensional boundary layer. However, we can observe that (20), which is

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the equation f o r a three-dimensional dis turbance i n a three-dimensional boundary

l aye r , is also a fourth-order equation.

more general cases t o (21).

equation. With W = 0 , t he transformation

This f a c t can be explo i ted t o relate

We w i l l illustrate t h i s p o s s i b i l i t y with Squi re ' s

reduces (20) to

- au"41 9

which is i d e n t i c a l t o (211, but i n t h e transformed var iab les .

unchanged, it is evident t h a t if a and B a r e real, a three-dimensional

s t a b i l i t y problem at Reynolds number R has been reduced t o a two-dimensional

problem a t t he lower Reynolds number G. theorem which states tha t i n a two-dimensional boundary l aye r with r e a l wave

numbers, i n s t a b i l i t y appears f i r s t f o r a two-dimensional dis turbance. Further-

more, i f ;= ;(;, 6) has been determined f o r a given U(y), then w = w(a, B , R)

is immediately known from (22).

Since U is

This is the ce lebra ted Squire

However, if a and 13 a r e complex o r the boundary l aye r is th ree dimensional,

t he u t : l i t y of t h e above transformation is lost.

complex, and i n t h e second t h e boundary-layer p r o f i l e is not i nva r i an t under

t h e transformation. I n both of these cases , there is l i t t l e point i n proceeding

beyond (20). The important conclusion is t h a t i n s t a b i l i t y problems governed by

In the f i r s t ins tance , is

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(7)-(10), the dc.termination of the eigenvalues only requires the solution of

a fourth-order equation.

4. Systcm of First-Order Equations

Since there are numerous stability problems that cannot be reduced to

a fourth-order -,ystem, a more flexible approach is to abandon the Orr-Somerfeld

equation altogather and work in terms of a system of first-order equations.

approach can hc: illustrated with (7)-(10) although it doesn't reveal its full

advantage until the eigenvalue problem is of higher than fourth order.

This

Let

af + @I = ii, ah - Bf = ah .

By adding and subtracting (7) and ( 9 ) , the following replacement equations can

be formed for these two linear combinations:

With,

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6 The f a c t tha'

confirms t h a t t h e eigenvalues can be obtained from a fourth-order system even

though we are r e a i l y deal ing nere Kith a sixth-order system. It is only t h e

determination of a l l of the eigenfunctions t h a t r equ i r e s t h e so lu t ion of the

f u l l sixth-order system.

complex and t o three-dimensional boundary layers .

Introduced only t o connect with other formulations, but has not been assigned a

meaning.

of wave propagation.

f i r s t four of t.lese equations do not contain Z or Z 5

This formulation is appl icable when a and 8 are

The quan t i ty 6 has been

When a and B are real, 6 is obviously t h e wave number in t h e d i r e c t i o n

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Ir. the freestream, (28)-(33) have constant coefficients and thus solutions

cf the form

%e characteristic walws occur in pairs, and are eas i ly determined to be

A = i ( ( r 2 4 B ) 2 112 , 1.2

'5.6 ' 3 , 4 '

(35)

(36)

(37)

where U and Wl are the freestream values of U(y) and W(y).

sign satisfies the boundary conditions at y -* -. for X are 1

Only the upper 1

The characteristic functions

2 2 li2 A':) = - i(a + 6 ) ,

= 1 , 3

- 12-

. . ~ ....

77-15

For real a, e and w this solution I s the linearized potential flow over a wavy

wall mviw in the oirecrlan of the wave number vector with weloclty w/(u + 6 2 2 112 . It can be called the lnvlecid solution, although this Interpretation is valid

only in the freestream. The characteristic functions for Xg are

*(3) P 1 3 ( 4 5 )

This solution represents a viscous wave and can be called the viscous solution.

The third solution is another viscous solution, and is

These three iinearly Independent solutions are the key to the numerical method

of obtaining eigenvalues as they provide the initial conditions of the numerical

integration.

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B. Temporal and S p a t l a l Theory

If a, and w are a l l real, t h e disturbance propagates through the

p a r a l l e l m a n flow w i t h constant rms amplitude. I f a and B are real, and w

is complex, the amplitude w i l l change with t i m e ; i f a and 8 are complex,

and is real, t h e amplitude sill change with x and z. The former case is

re fe r r ed t o as t he temporal amplif icat ion theory, t he la t ter as the s p a t i a l

amplif icat ion theory. I f a l l t h r e e q u a n t i t i e s are complex, t h e disturbance

w i l l grow i n both space and time. The o r i g i n a l , and f o r many years t h e only,

form of t he theory was t h e temporal theory.

the amplitude a t a f ixed point i s independent of t i m e and i t changes only

with distance.

d i r e c t manner than does the temporal theory.

1. Temporal Amplification Theory

However, i n a s teady mean flow

The s p a t i a l theory gives t h i s amplitude change i n a more

With w = wr + i w i and a and B r e a l , t he disturbance can be w r i t t e n

The magnitude of t h e wave number vector is

, G = ( a 2 + B ) 2 112 ,

and the angle between the d i r e c t i o n of a and t h e x a x i s is

The phase ve loc i ty , o r the v e l o c i t y w i t h which the c r e s t s move normal t o

themselves, is

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77-15

If A represenm the magnitude of q' ;t 8ome p a r t i c u l a r y, say t h e y f o r which

lq91 i s a marcfnrum, then it follows from (50) t h a t

We can i d e n t i f y wi as the temporal ampl i f ica t ion rate.

been chosen a t any y, and (541 would be t h e same. It is t h i s property t h a t

enables us t o t a l k ebout t h e "amplitude" of an i n s t a b i l i t y wave i n the 8-

manner as t he amplitude of a water wave even though t h i s amplitude is a funct ion

of y.

Obviously A could have

We may d i s t ingu i sh th ree poss ib le cases:

Mi 0 damped dis turbances,

wi - 0 n e u t r a l d i s turbances ,

wi 0 amplif ied dis turbances

( 5 5 )

The complex frequency may be writ ten

The real p a r t of c is equal t o t h e phase ve loc i ty c

ampl i f ica t ion rate.

s t a b i l i t y theory. However, It cannot be used i n t h e s p a t i a l theory, and s ince

wave theory usua l ly employs 6 and w, with t h e phase ve loc i ty be$ng introduced

as needed, we w i l l adopt t h c same procedure.

and arci is t h e temporal Ph'

The quant i ty c appears f requent ly i n the l i t e r a t u r e of

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77-15

2. S p a t i a l Amplification Theory

I n the s p a t i a l theory, w is real and the wave numbers i n the x and z

d i rec t ions are complex. With

a = a r + l a i , B = B r + i B i , (57)

w e can write t h e dis turbance i n the form

q'(x,y,z, t) = q(y) exp [-(alx + Biz)] exp f i (a r x + B I: z - u t ) ] . ( 5 8 )

By analogy with the temporal theory, we may define a real wave number by

- 2 2 112 a r = (ar + 6,)

The angle between the d i r ec t ion of Gr and the x axis is given by

-1 J, = t an (Br/ar) ,

and t h e phase ve loc i ty I s

c = Oli, Ph

(60)

A t t h i s po in t , i t I s tempting t o form a complex wave number 6 by (51)

with the real p a r t given by (59) and the Imaginary pa r t by a s imi l a r equation

in terms of ai and 6,. However, t h i s procedure is v a l i d only when

Bi/ai = $,/ar , (62)

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77-15

and it is not poss ib le t o make t h i 8 assumption and have the s p a t i a l theory

produce reasonable r e s u l t s .

ampl i f ica t ion from its o r i en ta t ion and introduce t h e new quan t i ty

Instead it is necessary to separa te t h e wave

- 2 2 1/2 a = (ai + Bi) i *

which makes an angle

with the x axis .

coordinate i n the d i r ec t ion of G,, we can rewrite (58) as

If x is t h e coordinate i n t h e d i r e c t i o n of Qr and E is t he

-L

It follows t h a t t he s p a t i a l ampl i f ica t ion rate is

- 1 - ( l / A ) (dA/dG) - a

To be more p rec i se , ( 6 6 ) gives the maximum s p a t i a l ampl i f ica t ion r a t e f o r the

p a r t i c u l a r choice of 5; t he re a r e l e s s e r ampl i f ica t ion r a t e s i n o ther

d i r ec t ions , and of cogrse 01

three-dimensional waves the s p a t i a l theory harl a d i f f i c u l t y not present i n t h e

temporal theory: i n addi t ion t o t h e wave o r i e n t a t i o n angle $, t h e maximum

ampl i f ica t ion d i r ec t ion 5 must be spec i f i ed before any ca l cu la t ions can be made.

- is i t s e l f a funct ion of $. We s e e t h a t f o r i

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The th ree cases which correspond t o ( 5 5 ) i n t he temparal theorv axe:

damped dis turbances,

n e u t r a l dis turbances,

amplified dis turbances.

3. Relation Between Temporal and Spa t i a l Theories

W e now have a temporal theory i n which t h e computation of t he eigen-

values is straj-ghtforward, but which does not y i e ld a s p a t i a l amplif icat ion

rate, and a s p a t i a l theory which y i e lds a s p a t i a l amplif icat ion rate, but only

a f t e r the unknown angle 5 has been specif ied. The problem of choosing 3 is

avoided only i n the spec ia l case of a two-dimerrsional dis turbance j n a two-

dimensional boundary l aye r where both $ and $ a r e zero.

both of these dilemmas is provided by introducing the powerful 2or.cept of

group ve loc i ty .

The r e so lu t ion of

A laminar boundary layer is a d ispers ive medium f o r t h e propagation of

i n s t a b i l j t y waves. That is, d i f f e r t v i frequencies propagate with d i f f e r e n t

phase v e l o c i t i e s , so t h a t the individual harmonic components i n a group of

waves a t one t i m e d i l l be dispersed (displaced) from each other a t some l a t e r

t i m e . An overa l l quant i ty , such as t h e energy densi ty or amplitude, does not

propagate with t h e phase ve loc i ty , but with t h e group ve loc i ty . Furthermore,

the group ve loc i ty can be ccnsidered a property of the individual waves, and

t o follow an individual frequency w e use t h e group ve loc i ty of t h a t frequency.

Consequently, an observer t r a v e l l i n g with the group ve loc i ty of a p a r t j

frequency w i l l always see tha t frequency and its associated amplitude. These

concepts were o r i g i n a l l y developed f o r f u l l y dispersed wave t r a i n s i n a homo-

, lar

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.

77-15

genaous medium with no d i s s ipa t ion ( r e a l a, 8 , w). However, i f t he inhomo-

genei ty of the F.?dium and the wave a t tenuat ion or amplificati t ,

over a wavelength, then t h e concepts s t i l l apply.

be made more prec ise by a multiple-scale perturbation ana lys i s .

appear t o be satisfied f o r Tollmien-Schlichrlng waves i n moderately iinstable

boundary layers . In addi t ion , an i n i t i a l a r b i t r a r y waveform quickly becomes

the dispersed wave t r a i n o t̂ the theory because, as shown by Mark '35) f o r the

temporal theory and by Corner, Houston and Ross '36) f o r t h e s p a t i a l theory, a l l

o ther modes except tb-. fundamental (Tollmien-Schlichting) mode are heavi ly

damped. The basic ideas of l i n e a r d i spars ive wave theory f o r conservative

systems are thoroughly discussed by Whitham. (37)

conservative boundary l aye r has been made by Landahl (38) (but see a l s o the

are both "small"

The meaning of "small" can

These condi t ions

e

An appl ica t ion t o t he non-

c r i t i c i s m by Stewartson ( 3 9 ) ) .

The dispers ion r e l a t i o n is

and t h e components of t h e (vector) ~ ''up ve loc i ty i n the x and z d i rec t ions

are obtained by d i f f e r e n t i s t i n g (68) with respect t o a and B .

w are real,

When a, B and

c g = ( $ , $ ) . ( 6 9 )

This same expression can be used i n the tempt :a1 theory with (0 replaced by

UJ

p a r t of the group ve loc i ty is neglected (and is zero a t the point of maximum

ampl i f ica t ion rate).

and i n the spa t ia l theory wi th ar and 8, f o r Q and 8 . The imaginary r '

6

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From what har a l r ead r been said, it is c l e a r t h a t the temporal and

s p a t i a l amplif icat ion rates are r e l a t e d by t h e group veloci ty . That is, from

the parallel-flow teqoral viewpoint we can form a s p a t i a l amplif icat ion rate

by following t h e wave with t k e group v e l o c i t y (now independent of x and 8 ) .

The time der€vative is thus transformed i n t o a space de r iva t ive by

d ' C - d

d t d i -

where must be i n the drrect ion of c Consequently, 8'

and the d i r e c t i o n of E, can be wr i t t en

The group ve loc i ty r e fe r r ed t o here is t h e group ve loc i ty of the temporal

theory.

The prcblem of converting a temporal t o a s p a t i a l amplif icat ion r a t e w a s

first encountered by S ~ h l i c h t i n g ' ~ ) , who used t h e two-dimensional vers ions

of (71) and (69) without comment. The same r e l a t i o n w a s a l s o used l a t e r by

Lees140', bu t t he f i r s t mathematical der ivat ion was given by Gaster ' 4 1 ) f o r

t he two-dimensional case.

simple and can e a s i l y be generalized t o th ree dimensions.

s t a r t s from the f a c t tha t the complex frequency w is an a n a l y t i c function of

t h e complex va r i ab le s a and 4 at a fixed x a n ' Therefore, the Cauchy-

Riemann equarions

The der ivat ion of Gaster's r e l a t i o n is q u i t e

The de r iva t ion

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77-15

aw aw, r - 0 -

aar aai

a W r aWi

wr a$, -E-

"i

' aai aar - I - -

(73)

can be applied. I n t h e two-dimensional case, the left-hand s i d e of t h e f e w s t

of these equations is t h e eroup ve loc i ty .

s i d e can be approximated by not ing t h a t w, decrease8 from Its temporal value

The de r iva t ive on t h e right-hand

to zero

theory.

l i n e a r ,

i n t h e s p a t i a l theory

I f t he amplif i c a t i o a

and

A

a s a

rate is small, these v a r i a t i o n s can b e considered

goes from zero t o its value i n t h e s p a t i a l i

W e see t h a t (74) is the same as t h e two-dimensional form of (71) w i t h

t h e important d i f f e rence t h a t t h e r e l a t i o n is revealed t o be only an approxi-

mation v a l i d f o r small ai.

arb1tiar:ly and the x a x i s ro t a t ed to l i e i n t h e $ cjlrection, (74) will sti l l

apply with ai replaced by ai and c

i n t he 5 d i rec t ion . Consequently, when

group ve loc i ty , (71) immediately follows, but again a s an approximation r a t h e r

than as an exact expression.

Ln the three-dimension& case, if J is spec i f i ed

by the component of t h e group ve loc i ty g

is chosen i n t h e d i r ec t ion of t he

C. Numerical Procedures

1. rn e8 of Methods

(3) Since the e a r l y 1960'8, t he asymptotic theory developed by Tollmien

and t in (4 ) has been l a rge ly superseded a s a means of producing numerical r e s u l t s

i n favor of direct so lu t ions of t h e governing d i f f e r e n t i a l equat ions on a

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77-15

d i g i t a l computer.

categories: (i) f i n i t e d i f f e rence methods, first employed by Thomas

h i s pioneering n m e r i c a l work of 1953; ( 2 ) shooting methods, first employed

by Brawn

as 1954); and (3) s p e c t r a l methods, f i r s t used by Gallagher and Mercer

with Chandrasekhar and Reid functions, and later improved by Orszag (44) w i t h

t he use of Chebyshev polynomials. A l l of these methods have advantages and

disadvantages which show up i n spec ia l i zed s i t u a t i o n s , but a l l are probably

equal ly a b l e t o do t h e rou t ine eigenvalue computations required i n t r a n s i t i o n

predict ion calculat ions.

been appl ied t o t h i s problem and w i l l be described here.

The numerical methods employed f a l l roughly i n t o th ree

(42) in

(a successful low Reynolds number program was operat ing as e a r l y

( 4 3 )

However, it is the shooting methods t h a t have mainly

( 4 5 ) After Brawn's i n i t i a l work, programs were developed by Mack,

( 4 7 ) Landahl and Kaplan, (16) Radbill and Van Driest, (46' Lee and Reynolds,

Wazzan, Okamura and Smith, (17) and Davey, (48) among others .

programs solve only the Orr-Sommerfeld eqliation; exceptfons are t h e

compressible program of Brown, (14) and the program of Mack ( 4 5 ) which was a l s o

o r i g i n a l l y developed f o r compressible flow and only l a t e r extended t o

incompressible flaw.

f ea tu re t h a t the numerical i n t eg ra t ion proceeds from t h e freestream t o t h e

w a l l .

Most of these

A l l of t he programs except Brown's have the common

The e a r l y app l i ca t ions of chocrting methods suffered from the. problem of

p a r a s i t i c e r r o r growth.

r ap id ly growing so lu t ion ( the l o c a l "viscous" so lu t ion ) which any numerical

This growth arises be:ause of t he presence of a

roundoff e r r o r w i l l follow. The r ap id ly growing e r r o r eventually completely

contaminates t h e less rap id ly growing so lu t ion .

coping with t h i s problem, which had previously l imited numerical so lu t ions t o

moderate Reynolds number, was made by Kaplan. (49)

The essentia; advance i n

The Kaplan method "ptirifies"

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77-15

t h e contaminated so lu t ion by f i l t e r i n g out t he p a r a s i t i c e r r o r whenever it

becomes l a r g e enough t o destroy linear independence.

2. Gram-Schmidt Orthonormalieat ion

An a l t e r n a t i v e method, f i r s t employed by Bellman and Kalaba (so) and

appl ied to t h e s t a b i l i t y problem by Radbi l l and Van Driest (46 ) and Wazean,

Okanarra and Smith, (17) is t h a t of Cram-Schmidt orthonormalization. This

method has t h e advantage t h a t i t is easier t o genera l ize t o higher-order

systems thaz is t h e Kaplan f i l t e r i n g technique. Eowever, t h e geometrical

argument o f t e n adduced in i ts support t h a t t h i s procedure preserves l i n e a r

independence by keeping the so lu t ion vec tors orthogonal cannot be co r rec t

because t h e so lu t ion vector space does not have a metric.

works on exac t ly the same b a s i s as Kaplan f i l t e r i n g :

replaced by a l i n e a r combination of t h e "small" and "large" so lu t ions which

is i t s e l f constrained to be "small."

Instead, t h e method

the "small*' s o l u t i o n is

For t h e simplest case of a two-dimensional wave i n a two-dimensional

boundaiy l aye r , t he re a r e two so lu t ions , 2 ( l ) and d3), each cons is t ing of four

components.

so lu t ion .

continues t o grow more rap id ly w i t h decreasing y than does 2").

p a r a s i t i c e r r o r w i l l follow Z(3', and when t h e d i f fe rence i n the magnitudes of

Z(3) and Z( l ) exceeds t h e computer word length, 2 ( l ) w i l l no lcnger be

independent of Z(3).

t i o n algorithm is applied.

In the freestream, Z(l) is t h e inv i sc id and Z(3) t he viscous

(3) Although t h i s i d e n t i f i c a t i o n is lost i n t he boundary l aye r , Z

The

Well before t h i s occurs, t he Gram-Schmidt orthonormaliza-

The "large" so lu t ion Z(') is normalized component

by component t o give t h e new so lu t ion

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77-15

where t h e a s t e r i s k r e f e r s t o a complex conjugate and {I to a s c a l a r product.

The s c a l a r product of 2(1) and S ( 3) is used t o form t he vector

to rep lace 2(')* where

bracket 8.

r e f e r s to. the quant i ty in t h e preceding square

(1) The numerical i n t eg ra t ion continues with Stl) and S(3) i n place of Z (3) i 5 and d3), and when in tu rn I S

single prec is ion a r i t hme t i c and a 36 b i t computer word, t h e or thonormali ta t ion

is repeated. With homogeneous boundary condi t ions a t t h e w a l l , i t makes no

d i f f e rence in t h e determination of t h e eigenvalues whether t h e Z ' s or S's are

used. A linear combination of the two so lu t ions s a t i s f i e s t h e f@) = 0

boundary condi t ion, but t he +(O) 5 0 boundary condi t ion w i l l i n genera l not

be s a t i s f i e d unless a, $ and w s a t i s f y an eigenvalue r e l a t ion .

3. Newton-Raphson Search Procedure

exceeds t h e set c r i t e r i o n o f , say 13 with

The Newton-Raphson method has been found to be s a t i s f a c t o r y f o r obtaining

t h e eigenvalues.

a

Because g(O), t h e t h i r d component of the l i n e a r combination of the two

independent so lu t ions Z(l) and Z(2) (or S ( l ) and S (2)) which s a t i s f i e s f(0) = 0 ,

is an a n a l y t i c funct ion of t h e complex va r i ab le a, even a f t e r orthonormalization,

tl a Cauchy-Riemann equat ions

In the s p a t i a l theory with w and $ f ixed , t h e guess value of

is perturbed by a small amount (A = 0.001 ar) an8 t h e in t eg ra t ion repeated. r

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77-15

can be appl ied t o e l imina te t h e need f o r a second in tegra t ion with ai

perturbed.

The cor rec t ions 6a and 6ai t o the i n i t i a l guesses ar and a r i are

obtained from t he r e s idua l O(0) and the numerical ( l i nea r ) approximations t o

the p a r t i a l der iva t ives by the equations

The corrected ar and ai are used t o start a new i t e r a t i o n , and the process

continued u n t i l 6a and 6a have been reduced below a prese t c r i t e r ion . r i

4. A Numerical Jbample

As an example we w i l l consider t h a t t he Reynolds number and frequency

are spec i f i ed f o r the f l a t -p l a t e boundary layer , and w e wish t o determine

both the complex wave number of the s p a t i a l theory and t h e wave number

and amplif icat ion rate of the temporal theory. It is convenient t o def ine

the length scale as

* * L = (v

With t h i s choice, the Reynolds number

* * 1/2 x /U1) .

I S

* * * * * * 112 = Re 1/2 . R = U1 L /v = (U1 x /v

(79)

The Reynolds numbers and wave numbers based on the displacement, momentum and

boundary-layer thicknesses a r e obtained by multiplying R and u based on L by *

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77--15

the appropriate dhens ionless thickness.

these fac tors are, i n the above order, 1.7208, 0.66411 and 6.0114 (u/U, - 0.999).

The usual form of the dimensionless frequency is

For the f la t -p la te boundary layer ,

* * *2 F = w u /U1 = w/R . (81)

For the example, w e choose

R = 1000, F = 0.3 x

and for the iaitial guess of the complex u i n the s p a t i a l theory,

u = 0.12 , ai = - 5.0 ; r

and for the temporal theory,

a = 0.12 , wi * 10.0 . r

The i n i t i a l conditions of the two independent so lu t ions are evaluated from

the formulas of Section 11-A.4, and the numerical integrat ion s t a r t ed a t

yl = y /L = 8.0. Each solut ion is integrated t o the wall ( a t y = 0 ) by

means of a fourth-order, f ixed s t ep s i ze , Runge-Kutta integrat ion. Other

integrat ion methods are a180 sa t i s fac tory , but t h i s simple method has been

found t o be trustworthy i n a wide bar ie ty of problems, par t icu lar ly i n t h e

d i f f i c u l t problem of determining the eigenvalue spectrum. ( 35)

s i z e in tegra tors are not recommended.

* *

Variable s tep-

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77-15

The operation of the eigenvalue search procedure is shown i n Table I.

The search is continued u n t i l 6 la I / la i is reduced below a p rese t c r i t e r i o n

(0.005 is usual ly adequate for three-place accuracy) i n t h e s p a t i a l theory,

and 6a/a and 61uil/iuil separa te ly s a t i s f y t h e c r i t e r i o n i n the temporal

theory.

i n t eg ra t ions are required per i t e r a t i o n ins tead of one as i n the s p a t i a l theory.

Two i t e r a t i o n s are usual ly s u f f i c i e n t t o achieve convergence i n a l a rge scale

computation where previously ca lcu la ted eigenv, ES can be used t o make good

i n i t i a l guesses.

Since (ilr is he ld constant i n the temporal theory, two per turha t ion

Poor i n i t i a l guesses were de l ibe ra t e ly chosen i n the examples.

Once a s i n g l e eigenvalue has been found f o r a given boundary layer , a l l

o the r s can be r ead i ly obtained.

gram t o produce a l l of t he unstable eigenvalues of a given frequency o r wave

number mesh up t o some spec i f i ed l a rge Reynolds number i n a s i n g l e computer run

Automatic procedures can be included i n the pro-

of a few minutes f o r a two-dimensional boundary layer .

t he i n i t i a l eigenvalue can prove troublesome.

inves t iga t ion is not f a r removed from one f o r which the eigenvalues are known,

i t is always poss ib le t o make a c lose enough guess f o r the Newton-Raphson

procedure t o converge. I f not , i t is necessary t o perform individual integra-

t i ons i n the complex a plane ( s p a t i a l ) o r w plane (temporal). Then the contour

The problem of obtaining

I f the boundary layer cinder

aqd Jordinson (51) or t h e more e labora te methods of Antar l i n e method of Mack, (35)

and Gaster, can be used t o I.ocate the i n i t i a l eigenvalue.

D. Some Numerical Resul ts

1. Amplification Proper t ies of Falkner-Skan P r o f i l e s

For a two-dimensional incompressible boundary layer on an impermeable

sur face of zero curvature and with no body force 0': surface hea t ing , the on ly

flow parameter l e f t is t h t pressure gradient . The e f f e c t on the amplif icat ion

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77-15

propert ies of systematic changes i n the pressure gradient can best be

demonstrated with the Falkner-Skan family of slmilar veloci ty prof i les .

The s ingle parameter of a Falkner-Skan boundary layer I s the dimcnslonless

pressure gradient

it is usual t o replace m by

(not to be confused with the lateral wave number). The values of B range from

1.0 f o r the two-dimensional stagnation point boundary :.ayer, through B = 0 f o r

the f la t p la te , t o B = 0.1988377 fo r the separation prof i le .

values of B represent reverse-f l o w p ro f i l e s . More negative

Two theore t ica l pr inciples are useful i n in te rpre t ing the numerical

resu l t s .

t ha t increasing the negative curvature of the veioci ty p ro f i l e near the wall

increases the s t a b i l i t y .

and states tha t as the inf lec t ion point moves away from the w a l l , the p r o f i l e

becoaes more unstable.

first e f f ec t occurs; as it decreases from zero, the second a f f ec t occurs.

The f i r s t comes from the viscous asymptotic theory, and states

The second comes from the inviscid theory of Rayleigh

We need only r e c a l l th.1.t as B Increases from zero, the

Some propert ies of Falkner-Skan p ro f i l e s a t e smmarized In Table 11,

wherr? a re l i s t e d 6 , 8 and 6 (the displacement, momentum and boundary-layer

th1cknesses)mde dimensionless with respect t o L ; H = 6 / O , the shape

factor ; U"(0) (= - m ) , t h e curvature a t the w a l l ; ys/6, the location of the

* * *

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77-15

i n f l e c t i o n point ; and cs, t h e dimensionless ve loc i ty at the i n f l e c t i o n point

and equal t o the phase speed of t he neu t r a l inv isc id disturbance.

The temporal s t a b i l i t y of Falher-Skan p r a f i l e s w a s computed i n grea t

( 8) d e t a i l by Pretsch

found i n Smith and Garnberoni (53)), and both the temporal and s p a t i a l s t a b i l i t y

from d i r e c t numerical so lu t ions by-Waeean, Okamura and Smith.

appl ica t ion t o the t r a n s i t i o n problem, the quant i ty of primary i n t e r e s t is

the i n t e g r a l of the s p a t i a l amplif icat ion rate f o r a constant frequency.

L still defined by (79), R by (80) and U1 by (82), it follows from (66) t h a t

from the asymptotic theory ( these r e s u l t s may also be

(17,181 For

With * *

where A. is the amplitude a t t h e i n i t i a l Reynolds number Ro.

t o take Ro as the lower-branch I Jtral s t a b i l i t y poin t ( i n i t i a l point of

i n s t a b i l i t y f o r t he frequency under consideration) i n order t o give a unique

meaning t o A/Ao.

(81) is t o be retained, then F is no longer constant f o r a constant dimensional

frequency, but is given by

It is convenient

I f the de f in i t i on of t he dimensionless frequency given by

* * * * Thus the ca lcu la t ion starts a t R2 - U (x ) x /v with dimensionless frequency

0 1 0 0 * * *2 * = w v /U1 (xo), and the in t eg ra l (84) is evaluated using a i ( R ) ca lcu la ted

FO

with the F(R) from (85).

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77-15

The length

a dimensionless. * i

* sca l e L is not the only possible choice with which t o make

* * The inverse of the u n i t Reynolds number, v /Ul , is of ten

found i n t he literature. (17 ,49) With t h i s c h o k e ,

+\ O I

1 E - -

* * - d R e ¶ (86)

where Re I s t he x-Reynolds number.

i t has the disadvantage t h a t aiv /U

even when the boundary l a y e r is unstable t o inv isc id disturbances.

contrary, the quant i ty a

can a l s o be used as the inv isc id amplif icat ion rate.

t o make between the viscous and inv isc id s t a b i l i t y theor ies in terms of

ai.

the viscous theory ( - a )

0.0202 a t the low Reynolds number of R = 400.

This procedure is per fec t ly acceptable, but * * *

(= a i /R) always goes t o zero as Re + - 1 On the

is based on a boundary-layer length scale, and so I

Comparisons are easy

For example, with f3 = - 0.15, the inv isc id (-a ) is 0.0199, and i n i max already reaches the s l i g h t l y higher value of i max

For the t ransl t ion-predict ion ca lcu la t ions of Section I V , th rce

quan t i t i e s w i l l be needed:

formed by the individual In (A/A ) vs. R curves; (ii) the frequency, Fmax ,

G C t t a envelope curve; and (3) a bandwidth AF of the frequency response

cui-v~s t o be discussed below. The envelope curves f o r sever71 values of B

are shm.3 i n Figure. 1.

pressure gradient , and the even s t ronger des t ab i l i z ing e f f e c t of an adverse

pressure gradient are c l ea r ly evident ,

shown i n Figure 2. (The

B * 0 curve and so cannot be shown.) We see a c l e a r d i s t i n c t i o n between

(i) the envelope curve, In (A/Ao)max vs. R ,

0

Both the s t rong s t a b i l i z i n g e f f e c t of a favorable

The corresponding frequencies are

= - 0.05 curve is almost coincident with the

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viscous and in f l ec t iona l i n s t ab i l i t y .

low frequency i n s t a b i l i t y , par t icu lar ly when the i n s t a b i l i t y only develops

at high Reynolds numbers. In contrast , i n f l ec t iona l i n s t a b i l i t y is a high

frequency i n s t a b i l i t y .

f o r B = -0.15 t o the corresponding frequency f o r B - 0.20 is 36.

Vfscous i n s t a b i l i t y is primarily a

The r a t i o of the frequency which gives In (A/Ao) = 9

The need f o r quantity (iii) .is shown i n Figure 3, where In (A/Ao) f o r

the f la t -p la te bctlndary layer is p lo t ted against F f o r several Reynolds

numbers.

width of unstable frequencies decreases.

layer response must be taken i n t o account i n any t r ans i t i on predict ion method

t h a t attempts t o calculate the integrated (over frequency) amplitude i n the

boundary layer.

bandwidth

As R increases, t he maximum amplitude r a t i o increases and the band-

This sharpening of the boundary-

A quant i ta t ive me6rswe of t h i s e f f e c t is provided by the

AF = Fmx - F[ln (A/Aolmax - 11 , (87)

where the second term is the frequency a t which A/Ao is l / e of (A/Ao)max on

t1.e low frequency s ide of Fmax.

ra ther than as a twQ-sided bandwidth because the p rac t i ca l requirements of

a large sca le eigenvalue calculat ion a r e such tha t A/Ao ran always be

obtained f o r frequencies smaller, not larger, than Fmx.

the r a t i o AF/Fmx a s a function of R f o r the same B as i n the previous

f igures . 2. Comparison of Temporal and Spat ia l Amplification Rates

It I s desirable t o define AF t h i s way

Figure 4 gives

The requirement t o know the direct ion of the group veloci ty before

computing eigenvalues from the s p a t i a l theory f o r other than two-dimensional

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waves i n two-dimensional boundary layers makes it worthwhile t o examine how

accurate are the s p a t i a l amplif icat ion rates obtained from the temporal

theory and Gaster's r e l a t i o n i n t h i s more r e s t r i c t e d case.

of the temporal theory, as given by (691, can be computed by numerical

d i f f e r e n t i a t i o n along with the other s t a b i l i t y proper t ies provided wr is a

smoothly varying function of the wave number.

The group ve loc i ty

Table I11 presents both mi and ai f o r four a ( three cimplified, including r t he maximum, and one damped) a t three d i f f e ren t B and a t Reynolds numbers with

l a rge emp'.ification rates. The case 6 = G provides amplif icat ion r a t e s t yp ica l

of boundary layers with small favorable pressure gradients ; B = - 0.10

provides about t he l a rges t amplif icatfon rates t h a t can be expected i n prac t ice ;

and the separat ion p r o f i l e c ~ i be viewed e? providing an upper l i m i t f o r

boundary-layer amplif icat ion rates, but na t pa r t i cu la r ly representa t ive of

ac tua l pract ice .

R -P

(The maximum ampiif icat ion r a t e of t h i s p r o f i l e occurs a t

and is -0.0480, which is only 12% la rge r than the R = 300 value of

-0.04?5. )

From Table 111 we see t h a t the Gaster r e l a t i o n is s a t i s f i e d q u i t e w e l l

a t B = 0, and less w e l l at the two o ther 13's. A t B = 0, the frequency and

phase ve loc i ty from the two theor ies also agree closely. It can a l s o be

noted t h a t t he r a t i o - wi/uy is less than the group ve loc i ty a t the maxiinum

amplti.icatior. rate f o r a l l th ree values of 6. The maximum di f fe rence between

c and -wi/aI f o r amplified disturbances occurs at the maxim*m amplif icat ion

r a t e , and is 1.62 f o r B = 0, 2.8% f o r B = -0.10, and 5.7% f o r the separat ion g

pro f i l e .

c between the t w o theor ies a t both l o w and high wave numbers. The concluston

can be drawn from Table I11 tha t the temporal theory and Gaster's r e l a t l o n

In the latter case, there are a l so important d i f fe rences i n wr and

Ph

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of 'er a s a t i s f a c t o r y method of obtaining spa t?a l a p l i f l c a t i o u r e t e s f o r

zero and favorable pressure gradients , but t h a t t h i s approach becomes

increasingly mere unre l iab le as the adverse pressure gradlent increases.

3. Group Velocity Direction f o r Oblique Waves

The importance of the choice of t he d i r ec t ion of maximum amplif icat ion

i n t h e s p a t i a l theory can be readf ly demonstrated by means of oblique waves

i n t h e f l a t -p l a t e boundary layer .

dimensional wave of dimensionless frequency F = 0.3 x

and f o r th ree oblique waves with JI = 45O, 60' and 75'.

and F f o r these waves were chosen so t h a t i n t h e JI d i rec t ion the Reynolds

number (R = Rcos J I ) and dimensionless frequency (P = F/cos* $1 are the same as

f o r t he two-dimensional wave. According t o (22), the Squire transformation,

t he s p a t i a l amplif icat ion rate i n the x-direction is

Table I V gives r e s u l t s for the two-

a t R = 1600,

The Reynolds numbers

ai - 6 , cos JI ,

where Gi is the s p a t i a l amplif icat ion rate of t he two-dimensional wave witb

F = 0.30 x

other th ree values should be -2.70, -1.91, and -0.989 f o r $ = 4S0, 60°, 75'

at R = 1600. Since Gi = -3.82 x from Table I V , the

respect ively. We can see t h a t , as already s t a t e d i n Section 11-A.2, this

r e l a t i o n I s t rue only when $ = JI. For 5 < I), a

by amounts ranging up t o 3 9 I . Consequently, it is not drmissible t o use

JI = y i n order t o preserve the real form of the Squire transformation.

Figure 5

is l a rge r than these values i

- The cor rec t 5 is the d i rec t ion of the group v- loci ty .

s how t h i s direction,now identl.@ed as $ and computed from t h e temporal

The important "h4Y theory, varies with J, f o r two f ixed values of 6 a t R = 1600.

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conclusion t o be drawn from Figure 5 is t h a t regard less of the o r i en ta t ion of

t he constant phase l i n e s ( the crests), the group l i n e s , which give the

f l i rect ion of energy propagation, remain concentrated near the freestream

direct ion.

the 8- as with 5 = 0 .

layer. it is a p ? r d s s i b l e approxinaation t o use t h e s p a t i a l theory w i t h $ = Oo.

H o w e v e r , it is still necessary t o use e i t h e r the complex form of the Squire

transformation u t the formulation given i n Section IX-A.4 to compute the

eigenvalues.

Furthermore, the loo en t ry of Table IV s h w s t h a t ai is nearly

Thus, f o r oblique waves i n a two-dimensional boundary

111. STABILITY EXPERIMR4TS

A. Schubauer-Skramstad Experiment

The l i n e a r s t a b i l i t y theory long went unappreciated except by its

founders because of the lack of any convincing experimental confirmation.

This needed conrumation w a s b r i l l i a n t l y supplied by the now classic experi-

ment of Schubauer and Skramstad") which was ca r r i ed out a t the National

Bureau of Standards i n the e a r l y 1943'9, but not published because of war-

time r e s t r i c t i o n s u t d i l 1948.

ingenuity of the experimenters and t o the development f o r the f i r s t time of a

wind tunnel with a r e a l l y low turbulence l eve l (about 0.03% i n the working

sect ion) .

anemometer placed i n the boundary layer of a f l a t p l a t e was displayed on an

osci l loscope screen, modulated s inusoidal wave t r a i n s with almost no random

This experiment owed its success both to the

When, with t h i s low turbulence level, the s igna l of a hot-wire

character were c l ea r ly seen.

t h a t these were t rue boundary-layer o s c i l l a t i o n s , and tha t they were the

cause, and not the e f f e c t , oi t rans i t ion .

Schubauer and Skramstad demonstrated conclusfvely

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In crder t o make a more quan t i t a t ive connection with the theory of

Tollmien and Schl icht ing, they used a v ib ra t ing ribbon i n order to produce

disturbances of a f ixed frequency with a control led inJ.t ia1 amplitude.

hot w i r e then measured the wave length, phsse ve loc i ty , and amplitude of

t h e a r t i f i c i a l l y p r o d u c e d waves as a function of Reynolds number.

The

Numerous

comparisons were made with the theory which were on the whole sa t i s f ac to ry ,

although the asymptotic theory a t t h a t time d i d not y i e ld very accurate

numerical r e su l t s .

experiment d id not l i e i n an exact quan t i t a t ive correspondence with theory,

However, i t is w e l l t o emphasize tha t the v i r t u e of t h i s

bu t r a t h e r in the systematic way t h a t a l l e s s e n t i a l fea tures of t he theory

were shown to be cor rec t , and i n the way the o s c i l l a t i o n s were shown t o

be necessary precursors of t rans i t ion .

The hot w i r e measures d i r e c t l y the rms disturban-e amplitude as a

function of downstream dis tance, and the amplif ic2t ion rates must be deduced

from the s lopes of such measurements.

measured amplitude has d i f f i c u l t i e s associated with non-parallel flow

e f f e c t s , i t is still of considerable i n t e r e s t t o compare the amplitude

Although the in t e rp re t a t ion of the

measurements with the quant i ty A of s t a b i l i t y theory. The comparison i s

given i n Figure 6 . The experimental wave amplitudes (u ' a t a f ixed d is tance

from the w a l l ) are a l l re fer red t o the amplitude a t xo = 2 in. behind the

ribbon which w a s located 4 A t . behind the leading edge of the p l a t e with

U1 = 64 f t l s e c .

(Ro = 1256), and A. is the amplitude a t xo rather than a t the neti tral

s t a b i l i t y point .

Therefore, the in tegra t ion of ai was a l so s t a r t e d a t x 0

The experimental points i n Figure 6 show conclusively tha t

the frequency is the fundamental parameter tha t determines whether a wave

w i l l be amplified o r damped, and the agreement with theory is sa t i s f ac to ry

although f a r from exact.

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This is a good place to br ing out some importa' j u a n t i t a t i v e aspec ts of

The r a t i o of the wavelength to t...: displacement thickness i n s t a b i l i t y waves.

i8

6 * a ( * x l , 2 ) & ~ e . r

A t R = 1256 (the xo of Figure 61, ar = 0.128 f o r the frequency with the

l a r g e s t amplif icat ion rate (f = 120 he, F = 0.311 x

(= 8.2 a), and we see t h a t unstable Tollmien-Schlichting waves are long

compared to the boundary-layer thickness.

assumes p a r a l l e l flow w a s one of t he o r i g i n a l c r i t i c i s m s made of t he theory,

--nd also explains why a considerable e f f o r t has been made recent ly t o develop

Another i n t e r e s t i n g point is the t d p i d i t y with non-parallel theories .

which the waves grow.

6 in. (= 88 6 , o r 3.1 A ) .

the f r a c t i o n a l change i n amplitude per wavelength, with ai and a

constant, is

* Hence A = 28.5 6

This r e s u l t from a theory t h a t

(32,33,54)

The 120 hz wave increases i n amplitude 2.5 times i n

From the d e f i n i t i o n of t?ie amplif icat ion rate, *

assumed r

o r about 33% f o r t he 120 hz wave. Although t h i s la rge growrh only e x i s t s

near the maximum amplif icat ion rate, i t still r a i s e s problems concerning

the appl ica t ion of kinematic wave concepts t o i n s t a b i l i t y waves.

E. Other Older Experiments

( 5 5 . 5 6 ) The next s t a b i l i t y ex3eriments were car r ied out by Liepmann.

These experiments were designed pr imari ly t o study the s t a b i l i t y and

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t r a n s i t i o n of boundary layers on concave surfaces , i.e. GSrt ler i n s t a b i l i t y ,

but measurements were also made on the convex s i d e of the curved p l a t e . Two

p l a t e s were used, with r a d i i of curvature of 20 f t . and 2-1/2 f t . , bu t only

the 20 f t . p l a t e w a s used f o r the de t a i l ed measurements on the convex s ide.

The boundary-layer ve loc i ty p r o f i l e was c lose t o t h a t of a f l a t p l a t e , but

the neu t r a l - s t ab i l i t y curve w a s found t o def ine an unstable region somewhat

l a r g e r than i n the Schubauer-Sttramstad expenment, perhaps r e f l e c t i n g a

s l i g h t average adverse pressure gradient .

convex s i d e s of both p l a t e s showed no e f f e c t of curvature, but the imposition

of favorable and adverse pressure grad ien ts on t he 20 f t . p l a t e d id produce

an e f f ec t .

decreased f o r an adverse pressure gradient , and the percentage change w a s

greater f o r t he favorable pre . 1- F gradient.

made with a pressure g r a d i e v

on the e f f e c t of pressure gra .LL

the experiments.

Transi t ion measurements on the

The t r a n s i t i o n Reynolds number was increased f o r a favorable and

No s t a b i l i t y measurements were

Indeed nc, t heo re t i ca l stat: i l i t y r e s u l t s

-ere ava i lab le t o Liepmann at the time of

The next experiment, by Bennett, ('') s tudied the inf luence of freestream

turbulence on t he i n s t a b i l i t y of the f l a t -p l a t e boundary layer .

i n s t a l l e d upstream of the test sec t ion t o raise the turbulence level t o

0.42%. The signal from a hot wire in t h e laroinar boundaq layer showed

f luc tua t ions of a random nature resembling turbulence w i t h l i t t l e evidence of

Tollnien-Sciillcht ing waves, an observation i n accord with Dryden ' s " ~ ) work

of 20 years previously.

was measured with a wave analyzer, a loca l peak w a s found t o develop at

about the most unstable frequency of l i n e a r s t a b i l i t y theory.

grew with increasing downstream dis tance , and gradually disappeared a f t e r

t r a n s i t i o n s t a r t e d as the spectrum evolved i n t o one typ ica l of a turbulent

A gr id was

However, when the p m e r spectrum of the f luc tua t ions

This peak

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boundary layer.

region t o determine the ariplitude h i s t o r y of the peak, and consequently no

conclusion is possible from t h i s experiment on t h e c r u c i a l point of whether

the linear disturbance growth is a f fec t ed by the external turbulence leve l .

Measurements were made at too few x s t a t i o n s i n t h e laminar

A flow v i sua l i za t ion technique f o r i n s t a b i l i t y waves using te l lur ium

coated rods w a s developed by Wortmann (”) f o r use i n a water boundary layer .

When a vol tage w a s applied t o the rods, t e l lu r ium ions, which are black, w e r e

re leased and made the i n s t a b i l i t y waves c l ea r ly v i s ib l e .

t o make quan t i t a t ive measurements, and a neu t r a l s t a b i l i t y curve w a s determined

t h a t compared favorably with Schubauer and Skramstad’s.

t i o n techniques t h a t have been used are smoke, (60s61) dye(62) and hyd -ogen

bubbles. (63)

waves, but have been appl ied mainly t o study the d e t a i l s of the t r a n s i t i o n

process following l i n e a r amplification.

It w a s then possible

Other flow visual iza-

These methods a l l show the presence of Tollmien-Schlichting

The hot-wire anemometer continued to be the primary t o o l of t h e NBS

t r a n s i t i o n s tudies , which, following the work of Schubauer and Skramstad, a l s o

concentrated on t h e non-linear region.

t he c h a r a c t e r i s t i c s of the turbulent spot ; Klebanoff and Tidstrom (65) and

Klebanoff, Tidstrom and Sargent (66) the sequence of events from the end of

t he l i n e a r region t o the f i r s t appearance of a turbulent spot. It was found

t h a t t he i n i t i a l l y two-dimensional Tollmien-Schlichting wave develops a span-

w i s e pe r iod ic i ty i n amplitude while i t is st ill undergoing ].inear amplif ica-

t ion.

since been observed i n o ther wind tunnels, but no convincing explanation has

(61) y e t been given f o r the o r ig in of t h i s per iodic i ty .

smoke p i c tu re s c l e a r l y show ths t a similar three-dimensionality also

develops i n na tu ra l t r ans i t i on .

Schubauer and Klebanoff (64) studied

There appears t 9 be a c h a r a c t e r i s t i c spanwise wavelength t h a t has

The Knapp and Roache

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C. Three Recent Experhents

1. ROSS, Barnes, Bums and Ross Fhperines

Uhat mrmts to a r epe t i t i on of t h a t p a r t of :he Schubauer-Skramstad

experiment which measured the i n s t a b i l i t y of a f l a t -p l a t e boundary layer w a s

ca r r i ed out by Ross, Barnes, Burns and Ross. (67)

same turbulence level as the NBS tunnel, w a s of s i m i l e r size, the f l a t p l a t e

w a s a l s o mounted v e r t i c a l l y , aud a v ibra t ing ribbon w a s used t o produce the

i n s t a b i l i t y waves.

ca lcu la t ions of Jordinson, (68) and exce l len t agreement w a s obtained for the

d i s t r i b u t i o n of u' through t h e boundary layer.

obtained f o r che A/Ao of th ree frequencies <F x lo4 = 0.82, 1.10, 1.571.

Pa r t i cu la r a t t e n t i o n w a s paid t o the region of the min5mum c r i t i c a l Reynolds

number and t h e maximum unstable frequency.

d i f f f c u l t because the boundary l aye r is both t h i n and rap id ly growing.

neu t r a l - s t ab i l i t y curve w a s a r r ived at with R = 2% and F = 4 x 10

compared t o the theo re t i ca l values of R = 302 ant ' F = 2.5 x 10 . The

d i f fe rences between theory and experiment w e r e a t t r i b u t e d t o non-parallel flow

e f f e c t s , a supposit ion s ince confirrmd by the good agreement of "Le Saric-

Nayfeh t h e ~ r y ( ~ ~ ) w i t h the above e v e r i m e n t a l r e s u l t s .

Thetr wind tunnel had the

"he hot-wfre measurements were compared with t h e de ta i led

Qu i t e good agreement w a s a l s o

Mecsurements i n t h i s region are

A

-4 c r P

4 cr P

It m u s t be pointed out t ha t the frequencies t h a t define the Rcr

port ion of t he neu t r a l - s t ab i l i t y curve a re not those tha t a r e important t o

t r a n s i t i o n i n environments w i c h small disturbances. For example, a t t h e

Schubauer-Skramstad t r a n s i t i o n Reynolds number of 2.8 x 10 , we see frori

= 2.;0 f igure 2 t h a t Fmnx 3.29 x compared t o F = 3.5 x 10 a t Rcr

according t o Sa r i c and Nayfeh. "lie former frequency first becones unstable

a t R = 820, where the non-parallel flow e f f e c t s a r e less severe :ban a t the

minimum c r i t i c a l Reynolds number. I h e non-parallel flow e f f e c t s w i l l be

6

-4

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even fu r the r reduced in boundary l aye r s with favorable pressure grad ien ts

where i n s t a b i l i t y occurs at much higher Reynolds numbers than f o r the f l a t

p la te .

2. Klebanoff-Tidstrom Experiment

It is surpr i s ing tha t tne t r a d i t i o n a l low-speed s t a b i l i t y theory has

never been t e s t ed experimentally f o r o ther than the Blasius boundary layer

s tudied by Schubauer and Skramstad.

experiments on t h e e f f e c t s of ro ta t ion , ( 6 9 ) compressibi l i ty ,

There have been boundary-layer s t a b i l i t y

f r e e (70)

convect ion, (71) and a heated w a l l i n water, '72) but t he accompanying

theo r i e s represented extensions of the e x i s t i n g theory t o new flow s i tua t ions .

There have been t r a n s i t i o n experiments on the technica l ly important e f f e c t s

of pressure gradient and suct ton, but no s t a b i l i t y experiments.

There is an important aspect of t r a n s i t i o n t h a t a l s o has received

l i t t l e a t t en t ion , and t h a t is the r e l a t ion of a p a r t i c u l a r disturbance

source t o the t r a n s i t i o n process i n a boundary layer . In o ther words, t h e

p rec ise mechanism by which, say, freestream turbulence, sound, and d i f f e ren t

types of roughness cause t r a n s i t i o n remains t o be discovered. Only i n the

case of two-dimensional roughness has the mechanism been found thanks t o a

remarkable experiment by ha. anoff and Tidstrom. (73)

elements with a diameter of about 0.8 6 , these inves t iga tors determined t h a t

For c i r c u l a r roughness

*

t r a n s i t i o n was moved forward from its normal locat ion, not by a disturbance

introduced i n t o the boundary layer by the roughness, but by t h e increased

amplif icat ion of an already ex i s t ing i n s t a b i l i t y wave i n the pressure

recovery zone behind the roughness. In a c e r t a i n sense, w2 do have here

an example of a s t a b i l i t y experiment with a pressure grad ien t , but t h i s

very spec ia l flow is scsrce ly representa t ive of pressure-gradient boundary

layers .

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Although t h e demonstration by Klebanoff and Tidstrom of the a c t u a l

t r a n s i t i o n mechanism w a s complete i n i t s e l f and d id not r e l y on t h e o r e t i c a l

comparisons, they d id compare t h e i r disturbance growth measurements with

r e s u l t s obtained from Pretsch 's char t s . ( 8 ) Unseparated adverse pressure

gradient Falkner-Skan p r o f i l e s w e r e f i t t e d t o t h e measured p r o f i l e s on the

bas i s of the shape f ac to r H.

good agreement. However, there are three objec t ions t h a t can be ra i sed

t o Klebanoff and Tidstrom's procedure: (i) t h e p r o f i l e s i n the i n i t i a l

p a r t of the recovery zone are c l e a r l y separated; (ii) the Pretsch c h a r t s

are not very accurate f o r adverse pressure gradients : and ( i i i ) the group,

and not t he phase, ve loc i ty should have been used t o transform temporal t o

s p a t i a l amplif icat ion L qtes.

The ca l cu la t ed and measured growths w e r e i n

When t h e computation is repeated with the cor rec t s p a t i a l ampl i f ica t ion

rates f o r the f i t t e d Falkner-Skan p r o f i l e s , t he r e s u l t s a r e not i n agreement

with the measurements. The obvious next s t e p is t o use the co r rec t ve loc i ty

p r o f i l e s which were measured i n grea t d e t a i l and with l i t t l e s c a t t e r .

Unfortunately, t he separated-flow region, which has a s t rong inf luence on

the i n s t a b i l i t y , could not be measured because of its closeness t o the wal l .

In any case, s t a b i l i t y ca lcu la t ions based on experimental curve f i t s t o

p r o f i l e s with an i n f l e c t i o n point a r e not l i k e l y t o be meaningful. The

conclusion t o be drawn is t h a t although the Klebanoff-Tidstrom experiment

f i rmly es tab l i shed t h a t two-dimensional roughness inf luences t r a n s i t i o n by

des t ab i l i z ing the boundary l aye r , t he amplif icat ion measurements cannot be

used as a test of a p a r t i c u l a r f o m of the s t a b i l i t y theory i n a rap id ly

varying pressure-gradient flow.

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1

3. Caster-Grant Experiment

Another recent s t a b i l i t y experiment of g rea t interest was ca r r i ed out

by Gaster and Grant (74) i n the same wind tunnel used i n the Ross, Barnes,

Burns and Ross experiment. A l l of the o the r s t a b i l i t y experiments with

a r t i f i c j a l l y produced disturbances have followed d i r e c t l y i n the Schubauer-

Skramstad t r a d i t i o n of deal ing with a Tollmien-Schlichting wave of a s i n g l e

frequency. This approach w a s followed even where the t r a d i t i o n a l v ib ra t ing

ribbon w a s replaced by a s i r e n ( 'O) o r a glow-discharge device. (75)

latter instance, oblique waves were a180 produced, but s t i l l only of a s i n g l e

frequency. Gaster and Grant used the completely d i f f e r e n t approach of trans-

mi t t ing an acoustic pulse from a loudspeaker through a s m a l l hole i n t h e i r

f l a t p l a t e t o produce a pulse dis turbance i n t h e boundary layer .

disturbance, after the rap id decay of a l l higher modes cons i s t s of Tollmien-

Schl icht ing waves of a l l frequencies and or ien te t ions . This type of experi-

ment, while not as s u i t e d t o mapping out s t a b i l i t y boundaries and making

the o ther usual checks of s t a b i l i t v theory as i n a v ibra t ing ribbon experiment,

is i n some respec ts c lose r t o t h e t r a n s i t i o n problem. An external disturbance,

such as freestream turbulence or sound, w i l l a l s o produce Tollmien-Schlichting

waves of a l l frequencies and or ien ta t ions , bu t in a random manner t h a t makes

the ensuing wave not ion i n the boundary layer d i f f i c u l t t o s o r t out.

pulse experiment gives P control led disturbance of t h i s type which enables

the d e t a i l s of a group of waves (a wave packet) , r a the r than a s ing le

Fourier component, t o be invest igated.

In the

This

The

I n the experiment, the hot w i r e was placed a t 15 spanwise s t a t i o n s

a t each of six downstream loca t ions t o record the passage of the wave

packet.

i n a shear layer have eppeared, (76-79) these papers a l l made use of asymptotic

Although severa l t heo re t i ca l treatments of the motion of a pulsc:

-42-

77-15

methods.

the wave packet w a s calculated from

I n order t o have mre exact numerical r e su l t s , the amplitude of

0 0

where x is the location of the pulse; and the Fourier coef f ic ien t has been

set equal t o uni~y.

e f f e c t of the boundary-layer growth, end the in tegra l of a(x) appears

€or t h i s same reason.

s p a t i a l s t a b i l i t y theory fo r a l l values of B and w by use of the Squire

transformation and a complex Reynolds number. In order t o evaluate the

double in t eg ra l with su f f i c i en t accuracy t o reproduce the wave packet,

10,900 eigenvalues were needed. The agreement thac w a s achieved

betweea theory and experiment fo r about the f i r s t 213 of the distance

covered w a s Impressive. After t h i s , the experimental wave packet

d i s tor ted i n a way noL given by the theory, possibly due t o non-linear

e f fec ts .

0

The fac tor x -'I4 is intended t o account f o r the

The eigenvalues a(x) were obtained from the

-4 3-

77-15

IV. TRANSITION PREDICTION

A. Nature of the Problem

A boundary layer haa a s p e c i f i c displacement thickness and sk in f r i c t i o n ,

but it does not have a spec i f i c t r a n s i t i o n Reynolds number. The observed transi-

t i o n Reynolds number dependo on the presence of disturbances i n the boundary

layer , which i n tu rn a r e r e l a t ed t o various d is twbance sources.

were no dis turbances, there would be no t r a n s i t i o n and the boundary layer would

remain laminar.

i n some way bringing i n the disturbances which cause i t , and any t r a n s i t i o n

c r i t e r i o n of an empirical nature can only be va l id f o r a very spec i f i c d i s -

turbance envirorrment .

I f there

Consequently, it is f u t i l e t o t a l k about t r a n s i t i o a without

Another point which must be s e t t l e d before going deeper i n t o the subjec t

is t o def ine the circumstances under which linear theory can be used to predic t

t r a n s i t i o n . Disturbances such as a large three-dimensional roughness element,

o r an air j e t , cause t r a n s i t i o n t o occur i n the immediate v i c i n i t y of t h e i r

location. However, i n many o ther instances, the disturbances act i n a more

ind i r ec t manner. A moderate freestream turbulence l eve l o r acous t ic i n t ens i ty

a f f e c t s t r a n s i t i o n by producing Tollmien-Schlichting waves i n the boundary layer

which then amplify, d i s t o r t , and f i n a l l y culminate i n the sudden appearance of

a turbulent spot . ( 6 4 )

linear theory, and i f the i n i t i a l disturbance amp1.itude is s u f f i c i e n t l y small,

the region of l i nea r growth can be of s ign i f i can t ex ten t . I n t h i s case, the

These i n s t a b i l i t y waves can be described i n i t i a l l y by

exponential growth of a l i n e a r dis turbance, and the absence of a n extensive

region of a i s t o r t e d laminar flow before the sudden breakdown t o turbulence

that is such a dis t inguishing fea ture of low-speed boundary-layer t r a n s i t i o n ,

makes it feas ib l e t o base a E t h o d of t r a n s i t i o n predict ion d i r e c t l y on the

linear theory i t s e l f .

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77-15

B. Amplitude Density Methods

1. Liepmann's Method

The f i r s t appl ica t ion of l i n e a r theory tr t r a n s i t i o n , by Schl icht ing, (4)

has already been meutioncd i n Section I.

Schl icht ing f o r appl ica t ion t? a i r f o i l s and described i n h i s book, (7) wae

A later method, developed by

based on the minimum c r f t i c a l Reynolds number, alld thus avoided having to

introduce the disturbance l e v e l .

a formula t h a t included mst of the ingredients needed t o vse linear theory

i n t r a n s i t i o n predict ion.

dens i ty method, because it considers only a s ing le frequency component, i . e . ,

a single s p e c t r a l l ine, fromwhat must be a contiramus dis turbance power

spectrum.

It was Lieprnann ( 5 6 ) who f i r s t worked out

This method can be c l a s s i f i e d as an amplitude

Liepmann's idea was t h a t t r a n s i t i o n should occur when the ratio of

the Reynolds stress of the boundary-layer disturbance equals the mean viscous

stress, and t h a t the l i n e a r theory provides

Reynolds stress. Liepmann's formula is

an adequate means of computing the

where cf is the sk in - f r i c t ion coe f f i c i en t .

Schl ich t ing (4'80) t o evaluate a l l terms i n (91) except the i n i t i a l

amplitude A . H e believed A. to be related t o the freestream turbulence

level, but the d e t a i l s of t h i s re la t ionship are as unknown today as they

were 30 years ago.

Liepmnn used results of

0

-45 -

77-15

9 2 . e Method

(53) The next s t e p was the independent development by Smith and Gamberoni

9 and by Van Ingen (81) of w h a t is of t en ca l l ed the e

s t a r t i n g from (91), found thac the only quant i ty on the right-hand s ide t h a t

they could compute other than the skin f r i c t i o n , namely the amplitude r a t i o

A/Ao, was i t s e l f s u f f i c i e n t t o correlate a large number of experimental mea-

surements of t r ans i t i on .

and an approximate method of c o q u t i n g the boundary layer developed by Smith,

t r a n s i t i o n w a s Bund t o occur when in(A/A ) * 9 .

by Van Ingen, bu t with the exponential f ac to r equal t o 7 o r 8 .

been j u s t l y c r i t i c i z e d f o r basing t r a n s i t i o n on a r a t i o and not on the d i s -

tuibance amplitude.

tunnel data which were used t o develop the method probably r e f e r t o r s t h e r

similar disturbance environments, and also doesn't give c r e d i t t o the value

of the e For a fixed value of Ao, which is

equivalent t o a fixed disturbance environment, the disturbance amplitude i n

the boundary layer does i n la rge measure vary as A/Ao.

e f f e c t on t r a n s i t i o n of changing a parameter which governs the mean boundary

l aye r , such as the p-&sure grad ien t , can i n t h i s pa r t i cu la r circumstance be

estimated by means of the s ing le fac tor A/Ao.

method. Smith and Gamberoni,

( 8 ) With the use of the s ta l . ' , l i ty charts of Pretsch,

(82)

A similar r e s u l t was obtained

This method has

0

However, t h i s c r i t i c i s m ignores the f a c t that the wind-

9 method i n comparative s tud ie s .

Consequently, the

I n more recent work, J a f f e , Okamra, and Smith (83) r?placed the o r ig ina l

approximate numerical methods of Smith and Gamberoni with more exact methods,

and Van 1nge11'~~) widened the range of appl ica t ions .

remains e s s e n t i a l l y as o r ig ina l ly developed, and the key to success s t i l l l ies

i n a judicious choice of the value of the exponential f ac to r . The use of the

method is s impl ic i ty i t s e l f once the necessary numerical t oo l s have been

However, the method

-46-

77-15

assembled. Three computer program are needed. The f i r s t computes the

inv isc id predsurr d i s t r i b u t i o n w e r a spec i f ied planar o r axisynrmetric

4 body shape; the second uses t h i s pressure d i s t r i b u t i m t o compute the laminar

boundary layer ; and the t h i r d computes the s p a t i a l amplif icat ion rate and

i t s i n t e g r a l , the amplitude r a t i o .

h ( A / A o ) reaches the chosen numerical f ac to r .

Trans i t ion is considered t o occur whenever

The . ~ . l y aspect of the method t h a t remains t o be mentioned is the

equation f o r A/Ao.

r e l a t i o n between the arc length x* and the boundary-layer thickness as d i th

With a non-similar boundary layer , there is no siwple

the Falkner-Skan p r o f i l e s . One convenient expression is .&.

(921

where c* is the chord 33: body length, U," i s the f reea t ream ve loc i ty , 'J,*(X*:

is the edge ve loc i ty , R8* is the loca l displacement thickness Re)nolds :lumber,

and (CY ) * = (Y * 6*. A s before , the i n t e g r d is evaluated fo r a constant i 8 i

dimensional frequency. Another expressicn, which preserves the boundary-

layer length scale L * * (x ) = [u* ~"/U,*(X*)]~, I s

* * where R,

3. Modified e Method

(11: x*/v*)' and Q, = CY L . i i

9

The main problem with the e9 method l ies i n the proper s e l ec t ion of

the exponential f a c t o r , which is by no means always equal t o nine.

t o avoid an a r b i t r a r y choice is t o relate the f ac to r , which can be ca l led n ,

One ..dy

-47-

77-15

t o the dbtUrDanc@ level i n an empirical manner.

applied t o the case of r'reestream turbulence by Van Ingen(84) and Mack.

#ind-tunm\, data on the influence of freestream turbulence on t r ans i t i on on

a f l a t p le te have been collected by Dryden(28) and are shown i n figure 7 .

A a r e recent a d -re complete col lect ion of data has been assembled by

Mall and GSbbixtgs, ('*) but the addftlonal data s t i l l follow the trend of

figure ?.

level is U ~ ' / U ~ , or T 5 [c,. 3 + 7 + 7 ) / 3 u1 I i f a l l three f luctuat ing

velocitv components have been masured. The turbulence level i n a low t u r -

bulence wind t u n m i l a increased by successively removing the damping screens,

and htgh levels are achieved by installing grid3 j u s t upstream of the test

tect ion.

turbulenze and sound.

t ransi t ion.

the s!gnal registered bj a hot-wire anemometer wfthout a f fec t ing the t r ans i t i on

Reynold8 nurihr. It is for t h i s reason that the cu~ve i n figure 7 is level

for T <O.lX.

by the s loping p; rtion of the curve €OK T > 0.1%.

This procedure has been

(85)

The start-of-tro. ,r i t ion Reynolds number is Re, and the turbulence

2 4

The total disturbance level i n a wind cunnel is made up of both

Below T = 0.1%, the sound component controls

As a r e s u l t , decreasing the turbulence component only decreases

The effect of freestream turbulence on t r ans i t i on is given

A n a n a l p i s of r e su l t s computed from the s t a b i l i t y theory shows that

t h i s p x t i o n of the curve can be accoucted for by considering A t o remain

fixed a t trar-altion, and with

2.4 A o - T . (94)

This same var ia t ion i n A/Ao 5.s given by l e t t i n g the exponential factor vary

according t o

n = - 8.43 - 2 . 4 kn T .

-4%-

I I

(95)

77-15

This mtbod , where n is re la ted d i r e c t l y to the disturbance l e v e l , may be

ca l led the modified e method. Whether it is v a l i d f o r o ther than f l a t -

p l a t e boundary layers cannot be determined u n t i l systematic t r a n s i t i o n da ta

comparable to f igure 7 become ava i l ab le f o r more general boundary layers.

9

C. Amplitude b t h o d

1. Formulation of amplitude Relation

Although the quant i ty A has been re fer red t o i n the foregoing as the

amplitude, it is more properly ca l l ed a n amplitude dens i ty because we have

been dealing so le ly with disturbances of a single frequency.

represents only a single s p e c t r a l l i n e of a continuous power spectrum.

method of Liepatann, the e’ and modified e

methods.

the energy dens i ty of the one-dimensional p m e r spectrum of the ex te rna l

disturbance w a s developed by Mack (89) for t ransi t ior . i n a supersonic wind

tunnel. Although t h i s method w a s reasonably successful i n explaining the

e f f e c t s of Mach number, u n i t Reynolds number and tunnel size on t r a n s i t i o n ,

i t can be c r i t i c i z e d on the bas i s that a s ing le s p e c t r a l l i n e is not an

adequate bas i s f o r r e l a t i n g the disturbance spectrum In the boundary layer

t o the spectrum of the ex terna l disturbance source.

That is, A

The 9 methods are a l l amplitude dens i ty

An amplitude dens i ty method which considers A t o be a function of

Consequently, another method has been developed which approximates

the ac tua l wide-band disturbance amplitude in the boundary layer .

point is the following expression for the disturbance amplitude Ad as a funct ion

The s t a r t i n g

of the Reynolds number R = (U1x/v) # :

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77-15

I n order t o c l a r i fy the der ivat ions, the use of a s t e r i sks has been dropped

and a l l quant i t ies are naw dimensional except A, A. and Ad which are referred

to the freestream veloci ty .

disturbance, t~ its (circular) frequency, and B its lateral wave number.

I n (96), A is the length scale of the external

The

sqitial disturbance i n the boundary layer is considered t o be produced d i r e c t l y

by the external disturbance and thus must be scaled i n terms of A, while the

amplitude ratio continues to be scaled i n terms of the boundary-layer l e r a th

scale. -- le basic assumption

in (96) is that a l l phase relations are randm so that the various harmonic

components add i n the square.

average of the disturbance amplitude, and we w i l l be determining the t r ans i t i on

The veloci ty sca le for both A. and A / A , is Ul.

Another viewpoint is that Ad is the lo- time

Reynolds number of t h i s average disturbance. What we would r e a l l y l i k e t o Know

is the average t r ans i t i on Reynolds number produced by a disturbance source tha t

is steady only i n its time average, and it is by no means ce r t a in that the two

Reynolds numbers are the same.

It is e n t i r e l y possible to evaluate the double in tegra l of (96) numeri-

ca l ly once A. is known, but it is more prudent t o adopt a simpler approach and

keep the numerical requirements nearly the same as fo r the e9 method. The two

l imiting cases which can be considered are where the bandwidth of the boundary-

layer response is small compared t o that of the external disturbance, a d where

the opposite prevai ls .

a c t s as a &function and (96) can be wr i t ten as

We w i l l treat only the f i r s t of these. Hence A/Ao

- 50-

77-15

W e reay fu r the r take advantage of che f a c t that the tesponsc cumes A/Ao

are c i t e n rather sbilar in shapc, and evaluate the ciouble in t eg ra l i n an

approxha te manner t o arrive at

The last two f ac to r s are cons i s t en t ly defined bandwidths of the frequency

and lateral wavenumber response curves.

from converting the in t eg ra t ion var iab les caA/U

layer var iab les m/U; and B(vx/U,)’. The c o m t a n t C expresses the d i f fe rence

between the exact and approximate in tegra t ions .

2

and 8A i n t o the boundary-

The f ac to r s (U,A/u) and 1 / R come

1

1

2 . In t e rac t ion Relation

In order t o proceed fu r the r i n the evaluat ion of Ad, it is necessary

to relate A t o the pa r t i cu la r e x t e r m l disturbance under considerat ion.

Unfortunately, a t the present time nothing is knawn about the mechanism by

which any ex te rna l disturbance produces Tollmien-Schlichting waves.

0

Although

we could proceed on a p x e l y empir ical b a s i s , a b e t t e r apprec ia t ion of the

problem is achieved by adopting a par t i cu la r viewpoint.

is that the i n s t a b i l i t y waves are produced i n the viscous sublayer , o r Stokes

layer , set up a t the w a l l by the freestream dis turbances. The forced response

i n t h i s layer t o a s inusoidal disturbance is obtained f r o m the s i n p l i f i e d model

of Prandt l . (*) i n studying the viscous

The viewpoint we adopt

This approach was used by Sternberg

sublayer of a turbulent boundary layer . The f i n a l s t e p is t o consider A t o

be proport ional t o the induced normal ve loc i ty of the forced response. The

0

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77-15

i d e a of relating the free t o t h e forced response had some success i n s tudying

supersonic boundary layers i r r a d i a t e d by strong a c o u s t i c d i s turbances . (91)

A s t ra ight forward a n a l y s i s leads to 9

w h e r e v; is t h e rms normal "Jiscous" v e l o c i t y of t h e harmonic component

( m n / v , , Bd, p ' is the corresponding rms pressure f l u c t u a t i o n a t t h e w a l l ,

and + is the d i r e c t i o n normal t o the cons tan t phase lines and cannot be near Q)

90'. The pressure f l u c t u a t i o n can be w r i t t e n more convenient ly as c

where p ' is the wide-band r m s pressure f l u c t u a t i o n , and i2 is a dimensionless

two-dimensional spectrum funct ion.

needed t o account f o r t h e d i s t r i b u t i o n of energy through d i f f e r e n t or ien-

t a t i o n s f o r t h e same frequency. What is being s a i d here is t h a t a harmonic

cmponent (cUn/v ,SA) of t h e f rees t ream dis turbance e x c i t e s a Tollmien-

S c h l i c h t i n g wave of t h e same frequency and o r i e n t a t i o n with an energy d e n s i t y

propor t iona l to t h e p r e s s u r e - f l u c t u a t i o n energy d e n s i t y of t h e freestream

component under t h e assumption t h a t t h e imposed pressure f l u c t u a t i o n a t the

w a l l is the same as i n t h e f rees t ream. The p r o p o r t i o n a l i t y f a c t o r is the

dimensionless frequency wv/Vl2, so t h a t t h i s l i n e of reasoning has produced

A two-dimensional spectrum funct ion is

1

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77-15

the r e s u l t that high frequencies are the mOst effecti-ve i n producing in s t a -

b i l i t y waves. There is no cross spectral t r ans fe r of entrgy.

D. Effect of Freestream Turbulence on Trans i t ion

1. Application of Amplitude Method

In order t o proceed fu r the r , we now restrict the freestream dis turbance

to turbulence, and furthermore consider the turbulence t o be i so t rop ic . A s

the pressure f luc tua t ion appearing i n (99) has been assumed to be the same as

i n the freestream, i so t rop ic turbulence theory (92) gives

Consequently ,

A s the x-Reynolds number does not appear i n (102), we may take A. t o be the

amplitude at the neu t r a l - s t ab i l i t y point .

For a band of waves of the same frequency and d i f f e r e n t o r i en ta t ions ,

the two-dimensional ( e = 0 ) component w i l l be the most amplified f o r f r e -

quencies along the envelooe curve of .Cn(A/Ao), i .e . , the maximam An(A/Ao) a t

a given Reynolds number.

Falkner-Skan p r o f i l e s .

i n addi t ion t o Ln(A/Ao)mx, F max and AF, involves the computation of four o r

f ive times as many eigenvalues as are otherwise needed.

r e s u l t s are ava i lab le t o make a general statement about the behavior of the

These frequencies are given i n f igure 2 for the

% The ca lcu la t ion of the la teral bandwidth AIB(wx/U1) 1

Not enough numerical

-53-

77-15

lateral bandwidth, but f o r the f l a t - p l a t e boundary layer i t decreases s l i g h t l y

wi th increasing R. For example, when defined, as ie AF, as the AB a t which :./Ao

has decreased t o l/e of its maximum value, it is 0.109 a t R = 900 m d 0.081 a t

R = 1600.

number, we w i l l proceed on the bas i s that the two-dimensional component i s

t h e mst amplified and that the lateral bandwidth is constant .

the subs t i t u t ion of (102) i n t o (98) y ie lds , with t h i s s impl i f ica t ion ,

Since t h i s change has l i t t l e influence on t h s t r a n s i t i o n Reynolds

Consequently,

4

A;(R) * C 2 ( + r ( 2 ) u A f F2 [(e) 1 x 1 max

(;2) maX A($ 1 max

The three quan t i t i e s on t he second l i n e of (103) are a l l determined from

s t a b i l i t y theory. The f i r s t two of these are needed f o r the e method, and

the t h i r d is determined as explained i n Section 1 1 - D . l without having t o do

any extra eigenvalue computations.

9

The quan t i t i e s i n the f i r s t l i n e of (103), except fo r R and the f r ee

It only remains t o develop constant C , are associated with the turbulence.

an expression f o r the spectrum function from i so t rop ic turbulence theory.

A convenient s t a r t i n g point i s von I & t d n ' s normalized ( t o 2n) in te rpola t ion

for the one-dimensional spectrum function of the longi tudinal ve loc i ty com-

ponent. The dimensionless frequency wh/Ul has been replaced by the dimensionless

- 54-

77-15

longitudinal wave number kl for the subsequent der ivat ion.

is permissible because a l l wave numbers kl have the convection veloci ty U1.

When Batchelor’s theory (92) is applied t o (104), a -713 ro l lo f f is obtained

for the one-dimensional pressure spectrum inczead of -5/3 as i n (104).

normaliLed interpolat ion formula equivalent t o von K a ’ d n ’ s which has t h i s

This procedure

A

behavior is

-7/6

Fl(kl) e 4 [1+(; k l f ]

The one-dimensional spectrum is given i n terms of the three-dimensional

spectrum by

Fz(kl) = 2n k Fj(k! dk , f kl

2 where k = (kl + k: + k;)’, and the two-dimensional spectrum is given i n

terms of the three-dimensional spectrum by

F2(k12) = F3(W dk3 ,

where k12 = (kl 2 + k2 2 # ) . Therefore,

dk .

An approximate evaluation of (lOS), which uses (105) and is correct for

and k12 = 0, is k12

F2(k12) = 1.78 [l + (0.82 k12 ;21-5/3

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77-15

It may be observed that the r o l l o f f i n F p is -10/3 as opposed t o -7/3 f o r

F1 and -S/3 f o r El .

by ml. It

may be observed that the in t eg ra l of r2 is equal t o F1(0)/2 as required by

the above equations.

For use i n (103) where k2( = 8) is zero, k12 Is replaced

The two spectrum functions F1 and F2 are s h w n i n f igure 8.

2. Numerical Results

A i l of the apparatus has now been assembled t o make use of (103), but

6 f i r s t the constant C must be evaluated.

a t T (or u;/U1) = 0.001 fo r the f l a t - p l a t e boundary layer , and wi th C = 1208

Ad = 0.04 a t this ReynGlds number f o r RA = UIA/v = 4 x LO . ca lcu la t ions , C retains t h i s value and t r a n s i t i o n is predicted t o start whenever

According t o f igure 7 , Ret = 2.8 x 10

4 I n a l l subsequent

Ad f i r s t reaches 0.04.

proportionate change i n C.

Any other value of Ad could have been chosen with a

It is easy to see from (103) that f o r given values of R and RE:, Ad

2 v a r i e s as T2, and Ad/T thus def ines a s ing le c u r .. ?ere are two e f f e c t s

of the scale Reynolds number R which tend t o oppose each o ther . The f i r s t

e f f e c t increases A

I!

through the appearance rf R as the propor t iona l i ty f a c t o r d A *

relating the bandwidths referred t o the boundary-layer sca le length t o the

bandwidths re fer red t o the turbulence length scale.

t h i s e f f e c t is that e e the turbvlence sca l e increases , the response curves

spread over a grhatr:: . ' r q ~ : n c y range of the turbulence power spectrum, with

the r e s u l t t h a t mor-? er f rgy is included i n the amplified band of frequencies

and the t r a n s i t i o n kynO1db number is reduced. The seco! . e f f e c t i s t h a t as

Another way t o look a t

2 1 R increases , the unstable frequency band, which is fixed i n terms of wv/U

(cf. f igure 3) , moves t o ' . igher dimensionless frequencies cuA/U, ( = F*RA)

with a smaller energy densi ty and the t r a n s i t i o n Reynolds number is increased.

r,

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77-15

The only reported measurements on the e f f e c t of scale on t r a n s i t i o n (") are

f o r turbulence levels between 1% and 3%, and show a marked increase i n the

t r a n s i t i o n Reynolds d e r with a n increase i n scale. The amplitude method

gives a decrease of 16% i n R e t a t T = 0.02% as Rr\ increases frcm 1 x lo4 to

8 x 10 and t h e f i r s t e f f e c t p reva i l s .

linear amplif icat ion region before t r a n s i t i o n , but t h i s region is not necessar i ly

w h a t is cont ro l l ing t r a n s i t i o n .

decreases as I$, goes from 1 X 10 to 2 x lo4 , but then increases by 169. as RA

increases fu r the r t o 8 x 10 and the second e f f e c t p reva i l s . This increasc

is smaller, and Ret h igher , than in t h e experiment.

4 A t T = 1%, the re is st i l l a subs t an t i a l

The Ret of the amplitude method i n i t i a l l y

4

4

Disturbance growth curves for the f l a t - p l a t e boundary layer are shown

i n f igure 9 f o r severa l turbulence levels.

curves which are the bas i c j u s t i f i c a t i o n f o r the use of a method based on

l i n e a r s t a b i l i t y theory f o r t r a n s i t i o n predict ion.

a function of T f o r s i x Falkner-Skan p r o f i l e s , and the experimental f l a t - p l a t e

da t a are repeated from f igure 7. The theo re t i ca l curve given by the amplitude

method f o r 8 = 0 agrees w e l l witl. these da ta fo r 0.09%

T < 0.09%, t r a n s i t i o n is cont ro l led by sound, and agreement with the theory

is not expected.

ment elsewhere, because the experimental da t a t o properly test the theory do

not e x i s t a t the present time.

possible to sort out the e f f e c t s of T and R w i l l i t be pcssible t o say whether

the method given above properly accounts fo r the e f f e c t of freestream turbulence.

The curves i n f igure 10 f o r the other values of 0 give some Idea of the

It is the s t eep s lopes of these

Figure 10 shows Ret as

T < 0.27%. For

There is l i t t l e poin t i n being concerned about lack of agree-

Only when enough da ta a r e ava i lab le t o make it

A

influence of a pressure gradient on t r a n s i t i o n .

f o r the pred ic t ion of t r a n s i t i o n i n a real boundary layer on the bas i s of l oca l

However, they cannct be used

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77-15

values of 8, because Ret depends on the amplif icat ion h i s t o r y of the dis turbance.

Only i f the p a r t i c u l a r value of B exists Over a subs t an t i a l port ion of the

boundary layer, a s i t u a t i o n most l i k e l y t o arise f r i small favorable pressure

grad ien ts , can f igure 10 be expected t o give a quan t i t a t ive estimate of the

pressure gradient e f f e c t .

e f f e c t s of 8 f o r several turbulence l eve l s . The e f f e c t of T on Re is almost

independent of B f o r adverse pressure grad ien ts , but f o r favorable pressure

grad ien ts T has a progressively reduced e f f e c t as 8 increases .

Figure 11 is a crossplot of the r e s u l t s t o show the

t

It is of i n t e r e s t to know the value a t t r a n s i t i o n of In(A/Ao)mx, 1.e.

9 the e f ac to r n, according t o the amplitude method.

course equal to nine f o r a l l boundary layers and a l l turbulence l eve l s ; i n the

modified e

Only with the amplitude method is n a function of both T and the boundary layer .

As $ decreases from B = 0 t o -0.1 wi th T = 0.1%, n decreases from 8.2 t o 7.7;

as 0 increases t o 0.2, n increases t o 10.1. The e f f e c t of t h i s va r i a t ion i n

n on Re is more c l e a r l y seen i f w e r e f e r t o the dashed curve i n f igure 11 of

the e method. This curve is c lose t o the T = 0.05% curve f o r B < -0.05, but

with higher values of 0 it depar t s more and more u n t i l a t 8 = 0 .2 it correrponds

to T

which vary by a f ac to r of four can a l l give n = 9 a t t r a n s i t i o n depending on

the value of 8 . Only experiments on pressure gradient boundary layers with

d i f f e r e n t turbulence leve ls can determine i f the t r a n s i t i o n Reynolds number

r e a l l y va r i e s with T and 0 as given by the amplitude method.

I n the e method, n is of

9 method n is given by (95) as a function of T f o r a l l boundary l aye r s .

t 9

0.7%. Thus the amplitude method gives the r e s u l t t h a t turbulence leve ls

The three xethods of t r a n s i t i o n predict ion which have been presented i n

t h i s sec t ion a l l use the same amount of computer time, because the eigenvalue

computations are what determine the time. Only if the l a t e r a l bandwidth fo r a

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77-15

two-dimensional boundary layer is determined . a t each Reynolds number does

the amplitude method require subs t an t i a l ly more t i m e than the o ther two

methods. The amplitude method does have the disadvantage t h a t a good

dea l must be known about the disturbance source, but t h i s is only a r e f l e c t i o n

of the. physical s i t u a t i o n t h a t t r a n s i t i o n is dependent on the type, i n t ens i ty

and spectrum of the disturbance source, and not j u s t on the boundary layer .

The method has been set up so that as information becomes ava i l ab le on the

mechanisms by which Tollmien-Schlicht ing waves are produced by ex te rna l d i s -

turbances, amre realistic l1,ceraction r e l a t ions can be e a s i l y incorporated

i n t o the computer program. It is hoped t h a t t he development of a method which

may f o r the f i r s t t i m e o f f e r t he p o s s i b i l i t y of a r a t i o n a l pred ic t ion of t rans-

a c t i o n K i l l encourage the experimental work which is so necesscry t o a r r i v e a t

this long sought goal.

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77-15

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M. Gaster and A. Davty, The Deveiopment of Three-Dimensional Wave-Packets i n Unbounded P a r a l l e l F l w s , J . Fluid Mech. 32, 801-808 (1968) . 8 . Schlicht ing, AmplitudemrerteIrung und Energiebilanz de r k le inen Stb'rungen b e i de r P l a t t e n s t r h n g , Gese l l s ch i f t de r Wissenschaften, G t t i n g e n , Math.- Phys. Klasse, Vol. 1, pp. 47-78 (1435).

3 . L. Van Ingcn, A Suggested Semi-Eqi r ica l Method f o r the Calculat ion of the Boundary La*pr Trans i t ion Region, Univ. of Technology, Dept. o f Aero. Eng. Report VTA-74, Delf t , Holland (1956).

A. M. 0 . Smith, Rapid Laminar Boundary-Layer Ca lcu la t ims by Piecewise Applicatton of Similar Solut ions, J. Aero. Sc i . 23, 901-912 (1956).

N. A. J a f f e , T. T. Okamura and A . M. 0 . Smith, Determination of Spa t i a l Amplification Factors and Their Application t o Trans i t ion , A M Jovrnal 8, 301-303 (1970) . J . L. Van Ingen, Trans i t ion , Pressure Gradient, Suction, Separation and S t a b i l i t y Theory, Law-Speed Boundary-Layer Trans i t ion Workshop: 11, Rand Corp., Santa Monica (1976) . L. PI. Mack, On the Effec t of Freestream Turbulence on Boundary-Layer Trans i t ion , Law-Speed Boundary-Layer Transi t ion Workshop: X I , Rand Cc Santa Monica, C a l i f . (1976).

A. A. Hal l and G . S. Hislop, Experiments on the Transi t ion of the Laminar Boundary Layer on a Z l a t P l a t e , Aero. Res. Comm. Reports and Memoranda 1843 (1938).

E. A. Wright and G. W . Bailey, Laminar F r i c t iona l Resistance with Pressure Gradient, J . Aero. Sc i . 5, 485-488 (1939).

D. J. Hall and J. C . Gibbings, Influence of Stream Turbulence and Pressure Gradient upon Boundary Layer Trans i t ion , J. Mech. Ew. Sci . l4, 134-146 (1972).

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89. L. M. Mack, A Numerical Method f o r t h e P r e d i c t i o n of High-speed Boundary- Layer T r a n s i t i o n U s i n g Linear Theory, Proceedings of Conf. on Aerodynamic Analyses Requiring Advanced Computers, NASA SP-347, pp. 101-124 (1975). NASA, Washington, D. C.

90. J . Sternberg, A Theory for the Viscous Sublayer of a Turbulent Flaw, J . Fluid He&. 13, 241-271 (1962).

91. L. M. m c k , Linear S t a b i l i t y Theory and t h e Problem of Supersonic Boundary- Layer T r a n s i t i o n , AIAA Journal l3, 278-289 (1975).

92. G. K. Batchelor, Pressure F luc tua t ions i n I s o t r o p i c Turbulence, m. Cambridpe P h i l . SOC. 47, 359-374 (1951).

T. von K&dn, Progress i n the Statistical Theory of Turbulence, J . Marine Research 1, 252-264 (1948).

93.

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SDat ia l Teawora 1

0 0.1200 -5.0

1 0.09421 -1.989

2 0.09918 -3.453

3 0.09964 -3.807

4 0.09964 -3.81 1

5 0.09964 -3.811

h - IW) I - e r - % I - Bix103 l m I

0.250 0.189 0.1200 10.0 0.250 0.259

C.3184 0.0774 0.1048 -3.319 0.2861 0.15?

0.3025 7 . 18x1(r3 0.1046 2.481 0.2869 0.0560

0.3011 5.29x1(r6 0.1001 1.223 0.2998 6.15x1r3

0.3011 7.88x1(r6 0.1001 1.410 0.2996 5 . 38x1(r0

0.3011 8 .83x1(r6 0.1001 1.412 0.2996 1 . 7 4 ~ 1 4 '

Table I. Operation of Eigenvalue Search Procedure

= 0, R = loon, F = O.3x1(r4, NSTEP = 80, yr = 8 , A = 0.001

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A

1 .o

0 .s

0.2

0.1

0 .os

0

-0 -025

-0 .(r5

-0.10

-0.15

-0.1988

* # 6 Re /x

0.6479

0.9854

i .32W

1.4891

1.5943

1.7208

1.7950

1.8790

2.0907

2.9149

3.4966

ate#/,

0.29234

0.42899

0.54770

0.60024

0.63044

0.66411

0.68254

0.70224

0.74637

0.79940

0.86811

bRef/x

3.1387

4.3207

5.2072

5.5733

5.7805

6.0114

6. i389

6.2771

6.5994

7.0385

8 -2382

H -

2 -2162

2.2969

2.4108

2.4809

2.5289

2 S911

2.6398

2.6758

2.8011

3.0209

4.0279

U " ( 0 )

- 1. OdOO

-0.33333

-0.11111

-0 .@5263

-0.02564

0

0.01235

3.02439

0.04762

0.06977

0.09043

-

0

0.1144

0.1647

0.2L12

0.3098

0.4216

C S -

-

0

0 -2178

0.2999

0 -4018

0 -4650

0.5026

Table 11. Properties of Falkner-Skan Boundary Layers

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f3 = 0, R = 1600

Spa t i a l Temporal

0.08 0.0215 -3.45% 0.269 0.0213 1.124 0.266 0.327 0.325

0.12 0.0351 -7.081 0.292 0.0350 2.499 0.292 0.359 0.353

0.16 0.0496 -2.890 0.310 0.0498 1.071 0.311 0.375 0.371

0.18 0.0578 3.566 0.321 0.0573 -1.345 0.318 0.369 0.377

J = -0.10, R = 700

0.07 0.0201 -5.730 0.1 ' 0.0196 2.060 0.280 0.358 0.360

0.13 0.0438 -13.71 0.337 0.0437 5.865 0.336 0.440 0.428

0.19 0.0719 -6.670 0 . 3 7 ~ 0.0726 3.532 0.382 0.537 0.530

3.25 0.1088 5.741 0.435 0.1081 -3.559 0.432 0.605 0.620

B = -0.1988, R = 300

0.05 0.0134 -16.64 0.268 0.0104 5.645 0.208 0.337 0.339

0.19 0.0784 -42.42 0.413 0.0782 22.90 0.412 0.571 0.540

0.29 0.1331 -28.96 0.459 0.1372 16.78 0.473 0.601 0.579

0.41 0.2161 17.65 0.527 0.2103 -11.17 0.513 0.618 0.633

Table 111. Comparison of Spa t ia l and Temporal Theories f o r Three Falkner-Skan

Boundary Layers

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R F x le si x 103 QI x io3 i

0" 1600 0.30 0" -3.82 -3.82

45" 2263 0.15 0" -3.17 -3.17

loo -3.12 -3.07

30" -3.32 -2.88

15O -3.82 -2.70

60' 320U 0.075 0" -2.46 -2.46

loo -2.42 -2.39

30" -2.59 -2 -24

45O -2.98 -2.11

60' -3.82 -1.91

75" 6182 0.0201 0" -1.37 -1.37

10" -1.36 -1.34

30" -1.49 -1.29

45" -1.75 -1.24

60' -2.32 -1.16

75O -3.82 -0.99

Table I V . Spa t ia l Amplification Rates of Oblique Waves i n the Plat-Plate Boundary Layer f o r Different Values of 8.

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Fiqure 1. Envelope curves of amDlitude ratio for Falkner-Skdn boundary lavers

191 a 6

4

2

I ,-I 8

0

4

Figure 2. Frequencv of amplitude-ratio en- velope curves for Falkner-Skan boundary 1 avcs r s

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F x lo4

Figure 3. Frequency-response curves of amplitude rat io at several Reynolds numbers for flat-plate boundary layer

Figure 4 . Revndlds number dependcnce of bandwidth of frequenLv-response curves for Falkncr-Skan boundary lavers

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3.0 1 f (hz) F x 10

4

2.5 120 0.311 - 2.0

- 1.5 a

1.0

0.5 40 0.104 -

m

80 0.207

160 0.415 - 60 0.156

180 0.466

-io

I

1 0

12 I 1 I I I I I

10 -

8 -

8 - 6 -

I*

4 -

2 -

I I 1 1 1 1 I

10 20 30 40 50 60 70 80

Figure 5. Direction of group velocity forobliquewaves of constant wave number in flat-plate boundary layer, Re112 = 1600

Figure 6. Comparison of theorywith Schubauer-Skramstad mea~urements(~) of the growth of six constant-frequency waves in flat-plate boundary layer

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? s a- X

aL

5

4

3 0

0

2 ,

1

0 . . - 0.02 0.04 0.1 0.2 3.4 1.0 2.0

T(%)

Figure 7. Effect of freestream turbulence on the transition Reynolds num- ber of the flat-plate boundary layer: o Schubauer-Skramstad ( 5 ) ; 0 Hall and Hislop(86); A Wright and Bailey(87); 0 Dryder~(~~); - modified e9 methnd

-74-

79-15

1.6 -

1.2 -

0.8 -

0.4 -

OO I

1

J ; I

3 4

= 1.78 [l + (0.82 k) 2 ] -5/3

5 4 7

k Figure 8. One- and two-dimensional Interpolation pressure )ectra of isotrnpic

turbulence. F1 i a normalized to 2n

-7 5-

77-15

5.0

1.5

4.0

%5

3. a

2.5 * - 0

.90

2. a

1.5

1.0

0.5

0

1 1 1 1 1 1

170.8 0.4 0.2 0.1 0.05 0.02%

Figure 9. Disturban-e amq1 itude growth i n f l a t - p l a t e boundary layer according t s amplitude meth-d for 1.everal freostrenm tur- bulence .l.evels, U l h / w = 4 - lo4

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77-15

‘igure 10. Effect of freestieam turbulence Oil the transition Reynolds number of Falkner-Skan hundary layers: U l M v = i x 104; 0, 0, A ? 0, experimental data (see caption of Figure ? for sources); - amplitude methor!

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6

4

2

lo7 8

6

4

a-

2

lo6 8

Q

4

2

1 o5 -0.15 -0.10 -0.05 0 0.05 0.10 0. 15 0.20

B Figure 11. Effect of pressure-gradLent parameter on the Lransition Re?-

nolds number of Falkner-Skan boundary l ayers accordin? t o smplitude method f w four levels of freestream t u i b u t m c e : Ulh/v 4 x 104; - amplitude method: - - - -. e9 metclod

-78- N A S I - 1PL - ten1 . I A , COI I